# Properties of distance spaces with power triangle inequalities

Daniel J. Greenhoe

► To cite this version:

Daniel J. Greenhoe. Properties of distance spaces with power triangle inequalities. Carpathian Mathematical Publications, 2016, 8 (1), pp.51-82. 10.15330/cmp.8.1.51-82. hal-01379725

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# Properties of distance spaces with power triangle inequalities

DANIEL J. GREENHOE

*Abstract:* *Metric spaces* provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the *triangle inequality*. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a *distance space*, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the *"power triangle inequality"*. The power triangle inequality represents an uncountably large class of inequalities, and includes the *triangle inequality*, *relaxed triangle inequality*, and *inframetric inequality* as special cases. The power triangle inequality is defined in terms of a function that is herein called the *power triangle function*. The power triangle function is itself a *power mean*, and as such is *continuous* and *monotone* with respect to its exponential parameter, and also includes the operations of *maximum*, *minimum*, *mean square*, *arithmetic mean*, *geometric mean*, and *harmonic mean* as special cases.

2010 Mathematics Subject Classification 54E25 (primary); 54A05,54A20 (secondary)

*Keywords:* metric space, distance space, semimetric space, quasi-metric space, triangle inequality, relaxed triangle inequality, inframetric, arithmetic mean, means square, geometric mean, harmonic mean, maximum, minimum, power mean

## Contents

1 **Introduction and summary** 2

2 **Standard definitions** 4

2.1 Standard sets 4

2.2 Relations 4

2.3 Set functions 4

2.4 Order 5

3 **Background: distance spaces** 5

3.1 Fundamental structure of distance spaces 6

3.1.1 Definitions 6

3.1.2 Properties 6

3.2 Open sets in distance spaces 6

3.2.1 Definitions 6

3.2.2 Properties 7

3.3 Sequences in distance spaces 9

3.3.1 Definitions 9

3.3.2 Properties 9

3.4 Examples 12

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<table><tr><td><b>4</b></td><td><b>Distance spaces with power triangle inequalities</b></td><td><b>15</b></td></tr><tr><td>4.1</td><td>Definitions</td><td>15</td></tr><tr><td>4.2</td><td>Properties</td><td>16</td></tr><tr><td>4.2.1</td><td>Relationships of the power triangle function</td><td>16</td></tr><tr><td>4.2.2</td><td>Properties of power distance spaces</td><td>17</td></tr><tr><td>4.3</td><td>Examples</td><td>24</td></tr><tr><td><b>A</b></td><td><b>Topological Spaces</b></td><td><b>26</b></td></tr><tr><td><b>B</b></td><td><b>Finite sums</b></td><td><b>30</b></td></tr><tr><td>B.1</td><td>Convexity</td><td>30</td></tr><tr><td>B.2</td><td>Power means</td><td>31</td></tr><tr><td>B.3</td><td>Inequalities</td><td>35</td></tr><tr><td><b>C</b></td><td><b>Metric preserving functions</b></td><td><b>35</b></td></tr><tr><td><b>Bibliography</b></td><td></td><td><b>37</b></td></tr><tr><td><b>Reference Index</b></td><td></td><td><b>44</b></td></tr><tr><td><b>Subject Index</b></td><td></td><td><b>45</b></td></tr></table>

# 1 Introduction and summary

*Metric spaces* provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the *triangle inequality*. However, there are several applications<fnref n="1" /> in which the triangle inequality does not hold but in which we would still like to perform analysis. So the questions that naturally follow are:

Q1. What happens if we remove the *triangle inequality* all together?

Q2. What happens if we replace the *triangle inequality* with a generalized relation?

A *distance space* is a *metric space* without the *triangle inequality* constraint. Section 3 introduces *distance spaces* and demonstrates that some properties commonly associated with *metric spaces* also hold in any *distance space*:

D1. Ø and X are *open* (Theorem 3.7 page 7)

D2. the intersection of a finite number of open sets is *open* (Theorem 3.7 page 7)

D3. the union of an arbitrary number of open sets is *open* (Theorem 3.7 page 7)

D4. every Cauchy sequence is *bounded* (Proposition 3.14 page 9)

D5. any subsequence of a *Cauchy sequence* is also *Cauchy* (Proposition 3.15 page 10)

D6. the *Cantor Intersection Theorem* holds (Theorem 3.18 page 11)

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<footnote n="1">¹references for applications in which the *triangle inequality* may not hold: [Maligranda and Orlicz(1987)] page 54 ("pseudonorm"), [Lin(1998)] ("similarity measures", Table 6), [Veltkamp and Hagedoorn(2000)] ("shape similarity measures"), [Veltkamp(2001)] ("shape matching"), [Costa et al.(2004)Costa, Castro, Rowstron, and Key] ("network distance estimation"), [Burstein et al.(2005)Burstein, Ulitsky, Tuller, and Chor] page 287 (distance matrices for "genome phylogenies"), [Jiménez and Yukich(2006)] page 224 ("statistical distances"), [Szirmai(2007)] page 388 ("geodesic ball"), [Crammer et al.(2007)Crammer, Kearns, and Wortman] page 326 ("decision-theoretic learning"), [Vitányi(2011)] page 2455 ("information distance"),</footnote>
The following five properties (M1–M5) *do* hold in any *metric space*. However, the examples from Section 3 listed below demonstrate that the five properties do *not* hold in all *distance spaces*:

<table><tr><td>M1. the metric function is continuous</td><td>fails to hold in</td><td>Example 3.21–Example 3.23</td></tr><tr><td>M2. open balls are open</td><td>fails to hold in</td><td>Example 3.21 and Example 3.22</td></tr><tr><td>M3. the open balls form a base for a topology</td><td>fails to hold in</td><td>Example 3.21 and Example 3.22</td></tr><tr><td>M4. the limits of convergent sequences are unique</td><td>fails to hold in</td><td>Example 3.21</td></tr><tr><td>M5. convergent sequences are Cauchy</td><td>fails to hold in</td><td>Example 3.22</td></tr></table>

Hence, Section 3 answers question Q1.

Section 4 begins to answer question Q2 by first introducing a new function, called the *power triangle function* in a *distance space* $(X, d)$, as $\tau(p, \sigma; x, y, z; d) \triangleq 2\sigma[\frac{1}{2}d^p(x, z) + \frac{1}{2}d^p(z, y)]^{\frac{1}{p}}$ for some $(p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^*.^2$ Section 4 then goes on to use this function to define a new relation, called the *power triangle inequality* in $(X, d)$, and defined as $\mathcal{A}(p, \sigma; d) \triangleq \{(x, y, z) \in \mathbb{X}^3 | d(x, y) \le \tau(p, \sigma; x, y, z; d)\}$.

The *power triangle inequality* is a generalized form of the *triangle inequality* in the sense that the two inequalities coincide at $(p, \sigma) = (1, 1)$. Other special values include $(1, \sigma)$ yielding the *relaxed triangle inequality* (and its associated *near metric space*) and $(\infty, \sigma)$ yielding the *$\sigma$-inframetric inequality* (and its associated *$\sigma$-inframetric space*). Collectively, a distance space with a power triangle inequality is herein called a *power distance space* and denoted $(X, d, p, \sigma)$.<fnref n="3" />

The *power triangle function*, at $\sigma = \frac{1}{2}$, is a special case of the *power mean* with $N = 2$ and $\lambda_1 = \lambda_2 = \frac{1}{2}$. *Power means* have the elegant properties of being *continuous* and *monontone* with respect to a free parameter $p$. From this it is easy to show that the *power triangle function* is also *continuous* and *monontone* with respect to both $p$ and $\sigma$. Special values of $p$ yield operators coinciding with *maximum*, *minimum*, *mean square*, *arithmetic mean*, *geometric mean*, and *harmonic mean*. *Power means* are briefly described in APPENDIX B.2.<fnref n="4" />

Section 4.2 investigates the properties of *power distance spaces*. In particular, it shows for what values of $(p, \sigma)$ the properties M1–M5 hold. Here is a summary of the results in a *power distance space* $(X, d, p, \sigma)$, for all $x, y, z \in X$:

(M1) holds for any $(p, \sigma) \in (\mathbb{R}^* \setminus \{0\}) \times \mathbb{R}^+$ such that $2\sigma = 2^{\frac{1}{p}}$ (Theorem 4.18 page 23)

(M2) holds for any $(p, \sigma) \in (\mathbb{R}^* \setminus \{0\}) \times \mathbb{R}^+$ such that $2\sigma \le 2^{\frac{1}{p}}$ (Corollary 4.14 page 21)

(M3) holds for any $(p, \sigma) \in (\mathbb{R}^* \setminus \{0\}) \times \mathbb{R}^+$ such that $2\sigma \le 2^{\frac{1}{p}}$ (Corollary 4.12 page 20)

(M4) holds for any $(p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+$ (Theorem 4.19 page 23)

(M5) holds for any $(p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+$ (Theorem 4.16 page 21)

APPENDIX A briefly introduces *topological spaces*. The *open balls* of any *metric space* form a *base* for a *topology*. This is largely due to the fact that in a *metric space*, *open balls* are *open*. Because of this, in *metric spaces* it is convenient to use topological structure to define and exploit analytic concepts such as *continuity*, *convergence*, *closed sets*, *closure*, *interior*, and *accumulation point*. For example, in a *metric space*, the traditional definition of defining continuity using *open balls* and the topological

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<footnote n="2">²where $\mathbb{R}^*$ is the set of extended real numbers and $\mathbb{R}^+$ is the set of positive real numbers (Definition 2.1 page 4)</footnote>

<footnote n="3">³ *power triangle inequality*: Definition 4.4 page 16; *power distance space*: Definition 4.3 page 16; examples of *power distance space*: Definition 4.5 page 16;</footnote>

<footnote n="4">⁴ *power triangle function*: Definition 4.1 (page 15); *power mean*: Definition B.6 (page 31); power mean is *continuous* and *monontone*: Theorem B.7 (page 31); power triangle function is *continuous* and *monontone*: Corollary 4.6 (page 16); Special values of $p$: Corollary 4.7 (page 17), Corollary B.8 (page 34)</footnote>
definition using open sets, coincide with each other. Again, this is largely because the open balls of a metric space are open.<fnref n="5" />

However, this is not the case for all *distance spaces*. In general, the open balls of a distance space are not open, and they are not a base for a topology. In fact, the open balls of a distance space are a base for a topology if and only if the open balls are open. While the open sets in a distance space do induce a topology, it's open balls may not.<fnref n="6" />

# 2 Standard definitions

## 2.1 Standard sets

**Definition 2.1** Let $\mathbb{R}$ be the **set of real numbers**. Let $\mathbb{R}^{+} \triangleq \{x \in \mathbb{R} | x \geq 0\}$ be the **set of non-negative real numbers**. Let $\mathbb{R}^{+} \triangleq \{x \in \mathbb{R} | x > 0\}$ be the **set of positive real numbers**. Let $\mathbb{R}^{*} \triangleq \mathbb{R} \cup \{-\infty, \infty\}$ be the set of **extended real numbers**. <fnref n="7" /> Let $\mathbb{Z}$ be the **set of integers**. Let $\mathbb{N} \triangleq \{n \in \mathbb{Z} | n \geq 1\}$ be the **set of natural numbers**. Let $\mathbb{Z}^{*} \triangleq \mathbb{Z} \cup \{-\infty, \infty\}$ be the extended set of integers.

**Definition 2.2** Let $X$ be a set. The quantity $2^X$ is the power set of $X$ such that
$$2^X \triangleq \{A \subseteq X\} \text{ (the set of all subsets of } X\text{).}$$

## 2.2 Relations

**Definition 2.3** <fnref n="8" /> Let $X$ and $Y$ be sets. The Cartesian product $X \times Y$ of $X$ and $Y$ is the set $X \times Y \triangleq \{(x, y) | x \in X \text{ and } y \in Y\}$. An ordered pair $(x, y)$ on $X$ and $Y$ is any element in $X \times Y$. A relation $\mathbin{\ Ring{\kern-0.5pt} \ Ring{\kern-0.5pt} }$ on $X$ and $Y$ is any subset of $X \times Y$ such that $\mathbin{\ Ring{\kern-0.5pt} \ Ring{\kern-0.5pt} } \subseteq X \times Y$. The set $2^{XY}$ is the **set of all relations** in $X \times Y$. A relation $\mathbin{\ Ring{\kern-0.5pt} \ Ring{\kern-0.5pt} } \in 2^{XY}$ is a **function** if $(x, y_1) \in \mathbin{\ Ring{\kern-0.5pt} \ Ring{\kern-0.5pt} }$ and $(x, y_2) \in \mathbin{\ Ring{\kern-0.5pt} \ Ring{\kern-0.5pt} } \implies y_1 = y_2$. The set $Y^X$ is the **set of all functions** in $2^{XY}$.

## 2.3 Set functions

**Definition 2.4** <fnref n="9" /> Let $2^X$ be the power set (Definition 2.2 page 4) of a set $X$.

A set $S(X)$ is a **set structure** on $X$ if $S(X) \subseteq 2^X$.

A set structure $Q(X)$ is a **paving** on $X$ if $\emptyset \in Q(X)$.

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<footnote n="5">⁵ *open ball*: Definition 3.5 page 6; *metric space*: Definition 4.5 page 16; *base*: Definition A.3 page 27; *topology*: Definition A.1 page 26; *open*: Definition 3.6 page 7; *continuity in topological space*: Definition A.11 page 28; *convergence in distance space*: Definition 3.11 page 9; *convergence in topological space*: Definition A.16 page 28; *closed set*: Definition A.1 page 26; *closure, interior, accumulation point*: Definition A.8 page 27; coincidence in all *metric spaces* and some *power distance spaces*: Theorem 4.15 page 21;</footnote>

<footnote n="6">⁶ *if and only if* statement: Theorem 3.10 page 8; open sets of a distance space induce a topology: Corollary 3.8 page 8;</footnote>

<footnote n="7">⁷ [Rana(2002)] pages 385–388 (Appendix A)</footnote>

<footnote n="8">⁸ [Maddux(2006)] page 4, [Halmos(1960)] pages 26–30, [Suppes(1972)] page 86, [Kelley(1955)] page 10, [Bourbaki(1939)], [Bottazzini(1986)] page 7, [Comtet(1974)] page 4 (|$Y^X$|); The notation $Y^X$ and $2^{XY}$ is motivated by the fact that for finite $X$ and $Y$, $|Y^X| = |Y|^{|X|}$ and $|2^{XY}| = 2^{|X|+|Y|}$.</footnote>

<footnote n="9">⁹ [Molchanov(2005)] page 389, [Pap(1995)] page 7, [Hahn and Rosenthal(1948)] page 254</footnote>
**Definition 2.5**<sup>10</sup> Let $\mathcal{Q}(X)$ be a paving (Definition 2.4 page 4) on a set $X$. Let $Y$ be a set containing the element 0. A function $m \in \mathcal{Y}^{\mathcal{Q}(X)}$ is a **set function** if $m(\emptyset) = 0$.

**Definition 2.6** The set function (Definition 2.5 page 5) $|A| \in \mathbb{Z}^{*2^X}$ is the cardinality of A such that

$$|A| \triangleq \begin{cases} \text{the number of elements in } A & \text{for finite } A \\ \infty & \text{otherwise} \end{cases} \quad \forall A \in \mathbb{2}^X$$

**Definition 2.7** Let $|X|$ be the cardinality (Definition 2.6 page 5) of a set $X$. The structure $\emptyset$ is the **empty set**, and is a set such that $|\emptyset| = 0$.

## 2.4 Order

**Definition 2.8**<sup>11</sup> Let $X$ be a set. A relation $\leq$ is an **order relation** in $\mathbb{2}^{XX}$ (Definition 2.3 page 4) if

1. $x \leq x$ and $y \leq z$ $\Rightarrow$ $x \leq z$ and $\forall x, y \in X$ (reflexive) and **preorder**

2. $x \leq y$ and $y \leq z$ $\Rightarrow$ $x \leq z$ and $\forall x, y \in X$ (transitive) and

3. $x \leq y$ and $y \leq x$ $\Rightarrow$ $x = y$ and $\forall x, y \in X$ (anti-symmetric)

An **ordered set** is the pair $(X, \leq)$.

**Definition 2.9**<sup>12</sup> In an ordered set $(X, \leq)$,

the set $[x : y] \triangleq \{z \in X | x \leq z \leq y\}$ is a **closed interval** and

the set $(x : y] \triangleq \{z \in X | x < z \leq y\}$ is a **half-open interval** and

the set $[x : y) \triangleq \{z \in X | x \leq z < y\}$ is a **half-open interval** and

the set $(x : y) \triangleq \{z \in X | x < z < y\}$ is an **open interval**.

**Definition 2.10** Let $(\mathbb{R}, \leq)$ be the ordered set of real numbers (Definition 2.8 page 5).

The absolute value $|\cdot| \in \mathbb{R}^\mathbb{R}$ is defined as<sup>13</sup>
$$|x| \triangleq \begin{cases} -x & \text{for } x \leq 0 \\ x & \text{otherwise} \end{cases}$$

# 3 Background: distance spaces

A distance space (Definition 3.1 page 6) can be defined as a metric space (Definition 4.5 page 16) without the triangle inequality constraint. Much of the material in this section about distance spaces is standard in metric spaces. However, this paper works through this material again to demonstrate “how far we can go”, and can't go, without the triangle inequality.

1. $|x| \geq 0$ and $x \in \mathbb{R}$ (non-negative) and

2. $|x| = 0 \Leftrightarrow x = 0$ and $x \in \mathbb{R}$ (nondegenerate) and

3. $|xy| = |x| \cdot |y|$ and $x, y \in \mathbb{R}$ (homogeneous / submultiplicative) and

4. $|x + y| \leq |x| + |y|$ and $x, y \in \mathbb{R}$ (subadditive / triangle inequality)

Reference: [Cohn(2002 December 6)] page 312

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<footnote n="10"><sup>10</sup> [Hahn and Rosenthal(1948)], [Choquet(1954)], [Pap(1995)] page 8 (Definition 2.3: extended real-valued set function), [Halmos(1950)] page 30 (§7. MEASURE ON RINGS),</footnote>

<footnote n="11"><sup>11</sup> [MacLane and Birkhoff(1999)] page 470, [Beran(1985)] page 1, [Korselt(1894)] page 156 (I, II, (1)), [Dedekind(1900)] page 373 (I–III). An order relation is also called a **partial order relation**. An ordered set is also called a **partially ordered set** or **poset**.</footnote>

<footnote n="12"><sup>12</sup> [Apostol(1975)] page 4, [Ore(1935)] page 409</footnote>

<footnote n="13"><sup>13</sup> A more general definition for absolute value is available for any commutative ring: Let $R$ be a commutative ring. A function $|\cdot|$ in $R^\mathbb{R}$ is an absolute value, or modulus, on $R$ if</footnote>
# 3.1 Fundamental structure of distance spaces

## 3.1.1 Definitions

**Definition 3.1**<sup>14</sup> A function $d$ in the set $\mathbb{R}^{X \times X}$ (Definition 2.3 page 4) is a **distance** if

$$1. \quad d(x, y) \ge 0 \qquad \forall x, y \in X \text{ (non-negative) and}$$

$$2. \quad d(x, y) = 0 \iff x = y \quad \forall x, y \in X \text{ (nondegenerate) and}$$

$$3. \quad d(x, y) = d(y, x) \qquad \forall x, y \in X \text{ (symmetric)}$$

The pair $(X, d)$ is a **distance space** if $d$ is a **distance** on a set $X$.

**Definition 3.2**<sup>15</sup> Let $(X, d)$ be a *distance space* and $2^X$ be the *power set* of $X$ (Definition 2.2 page 4). The **diameter** in $(X, d)$ of a set $A \in 2^X$ is $\text{diam } A \triangleq \begin{cases} 0 & \text{for } A = \emptyset \\ \sup \{d(x, y) | x, y \in A\} & \text{otherwise} \end{cases}$

**Definition 3.3**<sup>16</sup> Let $(X, d)$ be a *distance space*. Let $2^X$ be the *power set* (Definition 2.2 page 4) of $X$. A set $A$ is **bounded** in $(X, d)$ if $A \in 2^X$ and $\text{diam } A < \infty$.

## 3.1.2 Properties

**Remark 3.4** Let $(x_n)_{n \in \mathbb{Z}}$ be a *sequence* in a *distance space* $(X, d)$. The *distance space* $(X, d)$ does not necessarily have all the nice properties that a *metric space* (Definition 4.5 page 16) has. In particular, note the following:

1. $d$ is a *distance* in $(X, d) \iff d$ is continuous in $(X, d)$ (Example 3.23 page 14).

2. $B$ is an *open ball* in $(X, d) \iff B$ is open in $(X, d)$ (Example 3.22 page 13).

3. $B$ is the set of all *open balls* in $(X, d) \iff B$ is a base for a topology on $X$ (Example 3.22 page 13).<sup>17</sup>

4. $(x_n)$ is convergent in $(X, d) \iff$ limit is unique (Example 3.21 page 12).

5. $(x_n)$ is convergent in $(X, d) \iff (x_n)$ is *Cauchy* in $(X, d)$ (Example 3.22 page 13).

# 3.2 Open sets in distance spaces

## 3.2.1 Definitions

**Definition 3.5**<sup>18</sup> Let $(X, d)$ be a *distance space* (Definition 3.1 page 6). Let $\mathbb{R}^+$ be the *set of positive real numbers* (Definition 2.1 page 4).

An **open ball** centered at $x$ with radius $r$ is the set $B(x, r) \triangleq \{y \in X \mid d(x, y) < r\}$.

A **closed ball** centered at $x$ with radius $r$ is the set $\bar{B}(x, r) \triangleq \{y \in X \mid d(x, y) \le r\}$.

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<footnote n="14"><sup>14</sup> [Menger(1928)] page 76 ("Abstand *a* *b* definiert ist..." (distance from *a* to *b* is defined as...)), [Wilson(1931)] page 361 (§1., "distance", "semi-metric space"), [Blumenthal(1938)] page 38, [Blumenthal(1953)] page 7 ("DEFINITION 5.1. A distance space is called semimetric provided..."), [Galvin and Shore(1984)] page 67 ("distance function"), [Laos(1998)] page 118 ("distance space"), [Khamsi and Kirk(2001)] page 13 ("semimetric space"), [Bessenyei and Pales(2014)] page 2 ("semimetric space"), [Deza and Deza(2014)] page 3 ("distance (or dissimilarity)")</footnote>

<footnote n="15"><sup>15</sup> in *metric space*: [Hausdorff(1937)], page 166, [Copson(1968)], page 23, [Michel and Herget(1993)], page 267, [Molchanov(2005)] page 389</footnote>

<footnote n="16"><sup>16</sup> in *metric space*: [Thron(1966)], page 154 (definition 19.5), [Bruckner et al. (1997) Bruckner, Bruckner, and Thomson] page 356</footnote>

<footnote n="17"><sup>17</sup> [Heath(1961)] page 810 (THEOREM), [Galvin and Shore(1984)] page 71 (2.3 LEMMA)</footnote>

<footnote n="18"><sup>18</sup> in *metric space*: [Aliprantis and Burkinshaw(1998)], page 35</footnote>
**Definition 3.6** Let $(X, d)$ be a distance space. Let $X \setminus A$ be the set difference of $X$ and a set $A$. A set $U$ is open in $(X, d)$ if $U \in 2^X$ and for every $x$ in $U$ there exists $r \in \mathbb{R}^+$ such that $B(x, r) \subseteq U$. A set $U$ is an open set in $(X, d)$ if $U$ is open in $(X, d)$. A set $D$ is closed in $(X, d)$ if $(X \setminus D)$ is open. A set $D$ is a closed set in $(X, d)$ if $D$ is closed in $(X, d)$.

### 3.2.2 Properties

**Theorem 3.7**<sup>19</sup> Let $(X, d)$ be a DISTANCE SPACE. Let $N$ be any (finite) positive integer. Let $\Gamma$ be a SET possibly with an uncountable number of elements.

1. $X$ is OPEN.

2. $\emptyset$ is OPEN.

3. each element in $\{U_n|n=1,2,...,N\}$ is OPEN $\Rightarrow \bigcap_{n=1}^N U_n$ is OPEN.

4. each element in $\{U_\gamma \in 2^X | \gamma \in \Gamma\}$ is OPEN $\Rightarrow \bigcup_{\gamma \in \Gamma} U_\gamma$ is OPEN.

<figure><img src="image_14.png" /></figure>

PROOF:

(1) Proof that $X$ is open in $(X, d)$:

(a) By definition of open set (Definition 3.6 page 7), $X$ is open $\iff \forall x \in X \ \exists r$ such that $B(x, r) \subseteq X$.

(b) By definition of open ball (Definition 3.5 page 6), it is always true that $B(x, r) \subseteq X$ in $(X, d)$.

(c) Therefore, $X$ is open in $(X, d)$.

(2) Proof that $\emptyset$ is open in $(X, d)$:

(a) By definition of open set (Definition 3.6 page 7), $\emptyset$ is open $\iff \forall x \in X \ \exists r$ such that $B(x, r) \subseteq \emptyset$.

(b) By definition of empty set $\emptyset$ (Definition 2.7 page 5), this is always true because no $x$ is in $\emptyset$.

(c) Therefore, $\emptyset$ is open in $(X, d)$.

(3) Proof that $\bigcup U_\gamma$ is open in $(X, d)$:

(a) By definition of open set (Definition 3.6 page 7), $\bigcup U_\gamma$ is open $\iff \forall x \in \bigcup U_\gamma \ \exists r$ such that $B(x, r) \subseteq \bigcup U_\gamma$.

(b) If $x \in \bigcup U_\gamma$, then there is at least one $U \in \bigcup U_\gamma$ that contains $x$.

(c) By the left hypothesis in (4), that set $U$ is open and so for that $x, \exists r$ such that $B(x, r) \subseteq U \subseteq \bigcup U_\gamma$.

(d) Therefore, $\bigcup U_\gamma$ is open in $(X, d)$.

(4) Proof that $U_1$ and $U_2$ are open $\Rightarrow U_1 \cap U_2$ is open:

(a) By definition of open set (Definition 3.6 page 7), $U_1 \cap U_2$ is open $\iff \forall x \in U_1 \cap U_2 \ \exists r$ such that $B(x, r) \subseteq U_1 \cap U_2$.

(b) By the left hypothesis above, $U_1$ and $U_2$ are open; and by the definition of open sets (Definition 3.6 page 7), there exists $r_1$ and $r_2$ such that $B(x, r_1) \subseteq U_1$ and $B(x, r_2) \subseteq U_2$.

(c) Let $r = \min\{r_1, r_2\}$. Then $B(x, r) \subseteq U_1$ and $B(x, r) \subseteq U_2$.

(d) By definition of set intersection $\cap$ then, $B(x, r) \subseteq U_1 \cap U_2$.

(e) By definition of open set (Definition 3.6 page 7), $U_1 \cap U_2$ is open.

(5) Proof that $\bigcap_{n=1}^N U_n$ is open (by induction):

(a) Proof for $N=1$ case: $\bigcap_{n=1}^N U_n = \bigcap_{n=1}^1 U_n = U_1$ is open by hypothesis.

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<footnote n="19"><sup>19</sup> in *metric space*: [Dieudonné(1969)], pages 33–34, [Rosenlicht(1968)] page 39</footnote>
(b) Proof that $N$ case $\Rightarrow$ $N+1$ case:

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

**Corollary 3.8** Let $(X, d)$ be a DISTANCE SPACE. The set $T \triangleq \{U \in 2^X | U \text{ is OPEN in } (X, d)\}$ is a TOPOLOGY on $X$, and $(X, T)$ is a TOPOLOGICAL SPACE.

PROOF: This follows directly from the definition of an *open set* (Definition 3.6 page 7), Theorem 3.7 (page 7), and the definition of *topology* (Definition A.1 page 26).

<figure><img src="image_19.png" /></figure>

Of course it is possible to define a very large number of topologies even on a finite set with just a handful of elements;<fnref n="20" /> and it is possible to define an infinite number of topologies even on a *linearly ordered* infinite set like the *real line* ($\mathbb{R}$, $\leq$).<fnref n="21" /> Be that as it may, Definition 3.9 (next definition) defines a single but convenient *topological space* in terms of a *distance space*. Note that every *metric space* conveniently and naturally induces a *topological space* because the *open balls* of the metric space form a *base* for the *topology*. This is not the case for all distance spaces. But if the open balls of a *distance space* are all *open*, then those open balls induce a topology (next theorem).<fnref n="22" />

**Definition 3.9** Let $(X, d)$ be a *distance space*. The set $T \triangleq \{U \in 2^X | U \text{ is open in } (X, d)\}$ is the topology induced by $(X, d)$ on $X$. The pair $(X, T)$ is called the **topological space induced by** $(X, d)$.

For any *distance space* $(X, d)$, no matter how strange, there is guaranteed to be at least one *topological space induced by* $(X, d)$—and that is the *indiscrete topological space* (Example A.2 page 26) because for any distance space $(X, d)$, $\emptyset$ and $X$ are *open sets* in $(X, d)$ (Theorem 3.7 page 7).

**Theorem 3.10** Let $B$ be the set of all OPEN BALLS in a DISTANCE SPACE $(X, d)$. 
{ every OPEN BALL in $B$ is OPEN } $\Leftrightarrow$ 
{ $B$ is a BASE for a TOPOLOGY }

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<footnote n="20">²⁰ For a finite set $X$ with $n$ elements, there are 29 topologies on $X$ if $n = 3$, 6942 topologies on $X$ if $n = 5$, and 8,977,053,873,043 (almost 9 trillion) topologies on $X$ if $n = 10$. References: [ ] [oei(2014)] <http://oeis.org/A000798>, [ ] [Brown and Watson(1996)], page 31, [ ] [Comtet(1974)] page 229, [ ] [Comtet(1966)], [ ] [Chatterji(1967)], page 7, [ ] [Evans et al. (1967) Evans, Harary, and Lynn], [ ] [Krishnamurthy(1966)], page 157</footnote>

<footnote n="21">²¹For examples of topologies on the real line, see the following: [ ] [Adams and Franzosa(2008)] page 31 ("six topologies on the real line"), [ ] [Salzmann et al. (2007) Salzmann, Grundhöfer, Hähl, and Löwen] pages 64–70 (Weird topologies on the real line), [ ] [Murdeshwar(1990)] page 53 ("often used topologies on the real line"), [ ] [Joshi(1983)] pages 85–91 (§4.2 Examples of Topological Spaces)</footnote>

<footnote n="22">²² *metric space*: Definition 4.5 page 16; *open ball*: Definition 3.5 page 6; *base*: Definition A.3 page 27; *topology*: Definition A.1 page 26; not all open balls are open in a distance space: Example 3.21 (page 12) and Example 3.22 (page 13);</footnote>
<figure><img src="image_22.png" /></figure>

PROOF:

every open ball in **B** is open

$$ \Rightarrow \text{ for every } x \text{ in } B_y \in \mathbf{B} \text{ there exists } r \in \mathbb{R}^+ \text{ such that } B(x, r) \subseteq B_y \text{ by definition of open (Definition 3.6 page 7)} $$

$$ \Rightarrow \left\{ \begin{array}{l} \text{ for every } x \in X \text{ and for every } B_y \in \mathbf{B} \text{ containing } x, \\ \text{ there exists } B_x \in \mathbf{B} \text{ such that } x \in B_x \subseteq B_y. \end{array} \right\} \text{ because } \forall (x, r) \in X \times \mathbb{R}^+, B(x, r) \subseteq X $$

$$ \Rightarrow \mathbf{B} \text{ is a base for } \mathcal{T} \qquad \text{by Theorem A.4 page 27} $$

$$ \Rightarrow \left\{ \begin{array}{l} \text{ for every } x \in X \text{ and for every } U \subseteq T \text{ containing } x, \\ \text{ there exists } B_x \in \mathbf{B} \text{ such that } x \in B_x \subseteq U. \end{array} \right\} \text{ by Theorem A.4 page 27} $$

$$ \Rightarrow \left\{ \begin{array}{l} \text{ for every } x \in X \text{ and for every } B_y \in \mathbf{B} \subseteq T \text{ containing } x, \\ \text{ there exists } B_x \in \mathbf{B} \text{ such that } x \in B_x \subseteq B_y. \end{array} \right\} \text{ by definition of base (Definition A.3 page 27)} $$

$$ \Rightarrow \left\{ \begin{array}{l} \text{ for every } x \in B_y \in \mathbf{B} \subseteq T, \\ \text{ there exists } B_x \in \mathbf{B} \text{ such that } x \in B_x \subseteq B_y. \end{array} \right\} $$

$$ \Rightarrow \text{ every open ball in } \mathbf{B} \text{ is open} \qquad \text{by definition of open (Definition 3.6 page 7)} $$

<figure><img src="image_23.png" /></figure>

## 3.3 Sequences in distance spaces

### 3.3.1 Definitions

**Definition 3.11**<sup>23</sup> Let $(x_n \in X)_{n \in \mathbb{Z}}$ be a sequence in a distance space $(X, d)$. The sequence $(x_n)$ converges to a limit $x$ if for any $\varepsilon \in \mathbb{R}^+$, there exists $N \in \mathbb{Z}$ such that for all $n > N$, $d(x_n, x) < \varepsilon$. This condition can be expressed in any of the following forms:

1. The limit of the sequence $(x_n)$ is $x$.

3. $\lim_{n \to \infty} (x_n) = x$.

2. The sequence $(x_n)$ is convergent with limit $x$.

4. $(x_n) \to x$.

A sequence that converges is **convergent**.

**Definition 3.12**<sup>24</sup> Let $(x_n \in X)_{n \in \mathbb{Z}}$ be a sequence in a distance space $(X, d)$.

The sequence $(x_n)$ is a **Cauchy sequence** in $(X, d)$ if

for every $\varepsilon \in \mathbb{R}^+$, there exists $N \in \mathbb{Z}$ such that $\forall n, m > N$, $d(x_n, x_m) < \varepsilon$ (Cauchy condition).

**Definition 3.13**<sup>25</sup> Let $(x_n \in X)_{n \in \mathbb{Z}}$ be a sequence in a distance space $(X, d)$.

The sequence $(x_n \in X)_{n \in \mathbb{Z}}$ is **complete** in $(X, d)$ if

$(x_n)$ is **Cauchy** in $(X, d) \quad \Rightarrow \quad (x_n)$ is **convergent** in $(X, d)$.

### 3.3.2 Properties

**Proposition 3.14**<sup>26</sup> Let $(x_n \in X)_{n \in \mathbb{Z}}$ be a sequence in a distance space $(X, d)$.

$$ \{ (x_n) \text{ is CAUCHY in } (X, d) \} \Rightarrow \{ (x_n) \text{ is BOUNDED in } (X, d) \} $$

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<footnote n="23"><sup>23</sup> in *metric space*: ⇄ [Rosenlicht(1968)] page 45, ⇄ [Giles(1987)] page 37 (3.2 Definition), ⇄ [Khamsi and Kirk(2001)] page 13 (Definition 2.1) “→” symbol: ⇄ [Leathem(1905)] page 13 (section III.11)</footnote>

<footnote n="24"><sup>24</sup> in *metric space*: ⇄ [Apostol(1975)] page 73 (4.7), ⇄ [Rosenlicht(1968)] page 51</footnote>

<footnote n="25"><sup>25</sup> in *metric space*: ⇄ [Rosenlicht(1968)] page 52</footnote>

<footnote n="26"><sup>26</sup> in *metric space*: ⇄ [Giles(1987)] page 49 (Theorem 3.30)</footnote>
PROOF:

($x_n$) is *Cauchy* $\Rightarrow$ for every $\varepsilon \in \mathbb{R}^+$, $\exists N \in \mathbb{Z}$ such that $\forall n, m > N$, $d(x_n, x_m) < \varepsilon$ (by Definition 3.12 page 9)
$\Rightarrow \exists N \in \mathbb{Z}$ such that $\forall n, m > N$, $d(x_n, x_m) < 1$ (arbitrarily choose $\varepsilon \triangleq 1$)
$\Rightarrow \exists N \in \mathbb{Z}$ such that $\forall n, m \in \mathbb{Z}$, $d(x_n, x_{m+1}) < \max \{\{1\} \cup \{d(x_p, x_q) \mid p, q \not> N\}\}$
$\Rightarrow (x_n)$ is *bounded* (by Definition 3.3 page 6)

<figure><img src="image_26.png" /></figure>

**Proposition 3.15**<sup>27</sup> Let $(x_n)_{n \in \mathbb{Z}}$ be a sequence in a distance space $(X, d)$. Let $f \in \mathbb{Z}^{\mathbb{Z}}$ (Definition 2.3 page 4) be a strictly monotone function such that $f(n) < f(n+1)$.

$\underbrace{(x_n)_{n \in \mathbb{Z}} \text{ is CAUCHY}}_{\text{sequence is CAUCHY}} \Rightarrow \underbrace{(\underbrace{(x_{f(n)})}_{\text{subsequence is also CAUCHY}})_{n \in \mathbb{Z}} \text{ is CAUCHY}}$

PROOF:

($x_n$)$_{n \in \mathbb{Z}}$ is *Cauchy*
$\Rightarrow$ for any given $\varepsilon > 0$, $\exists N$ such that $\forall n, m > N$, $d(x_n, x_m) < \varepsilon$ by Definition 3.12 page 9
$\Rightarrow$ for any given $\varepsilon > 0$, $\exists N'$ such that $\forall f(n), f(m) > N'$, $d(x_{f(n)}, x_{f(m)}) < \varepsilon$
$\Rightarrow$ $(x_{f(n)})_{n \in \mathbb{Z}}$ is *Cauchy* by Definition 3.12 page 9

<figure><img src="image_27.png" /></figure>

**Theorem 3.16**<sup>28</sup> Let $(X, d)$ be a distance space. Let $A^-$ be the closure (Definition A.8 page 27) of $A$ in a topological space induced by $(X, d)$.

$$
\left\{
\begin{array}{ll}
1. & \text{LIMITS are UNIQUE in } (X, d) \quad \text{(Definition 3.11 page 9)} \quad \text{and} \\
2. & \text{(A, d) is COMPLETE in } (X, d) \quad \text{(Definition 3.13 page 9)}
\end{array}
\right\}
\quad
\Rightarrow
\quad
\underbrace{A \text{ is CLOSED in } (X, d)}_{A = A^{-}}
$$

PROOF:

(1) Proof that $A \subseteq A^{-}$: by Lemma A.10 page 27

(2) Proof that $A^{-} \subseteq A$ (proof that $x \in A^{-} \Rightarrow x \in A$):
    (a) Let $x$ be a point in $A^{-}$ ($x \in A^{-}$).
    (b) Define a *sequence* of open balls $((B(x, \frac{1}{2}), B(x, \frac{1}{2}), B(x, \frac{1}{3}), \dots))$.
    (c) Define a *sequence* of points $(x_1, x_2, x_3, \dots)$ such that $x_n \in B(x_n, \frac{1}{n}) \cap A$.
    (d) Then $(x_n)$ is *convergent* in $X$ with limit $x$ by Definition 3.11 page 9
    (e) and $(x_n)$ is *Cauchy* in $A$ by Definition 3.12 page 9.
    (f) By the hypothesis 2, $(x_n)$ is therefore also *convergent* in $A$. Let this limit be $y$. Note that $y \in A$.
    (g) By hypothesis 1, limits are *unique*, so $y = x$.
    (h) Because $y \in A$ (item (2f)) and $y = x$ (item (2g)), so $x \in A$.
    (i) Therefore, $x \in A^{-} \Rightarrow x \in A$ and $A^{-} \subseteq A$.

<figure><img src="image_28.png" /></figure>

**Proposition 3.17**<sup>29</sup> Let $(x_n)_{n \in \mathbb{Z}}$ be a sequence in a distance space $(X, d)$. Let $f: \mathbb{Z} \to \mathbb{Z}$ be a strictly

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

<footnote n="27"><sup>27</sup> in *metric space*: [Rosenlicht(1968)] page 52</footnote>

<footnote n="28"><sup>28</sup> in *metric space*: [Kubrusly(2001)] page 128 (Theorem 3.40), [Haaser and Sullivan(1991)] page 75 (6·10, 6·11 Propositions), [Bryant(1985)] page 40 (Theorem 3.6, 3.7), [Sutherland(1975)] pages 123–124</footnote>

<footnote n="29"><sup>29</sup> in *metric space*: [Rosenlicht(1968)] page 46</footnote>
increasing function such that $f(n) < f(n + 1)$.

$$ \underbrace{(x_n)_{n \in \mathbb{Z}} \to x}_{\text{sequence converges to limit } x} \implies \underbrace{(x_{f(n)})_{n \in \mathbb{Z}} \to x}_{\text{subsequence converges to the same limit } x} $$

<figure><img src="image_31.png" /></figure>

PROOF:

$$ (x_n)_{n \in \mathbb{Z}} \to x \implies \forall \varepsilon > 0, \exists N \text{ such that } \forall n > N, d(x_n, x) < \varepsilon \quad \text{by Theorem 4.15 page 21} $$

$$ \implies \forall \varepsilon > 0, \exists f(N) \text{ such that } \forall f(n) > f(N), d(x_{f(n)}, x) < \varepsilon $$

$$ \implies (x_{f(n)})_{n \in \mathbb{Z}} \to x \qquad \text{by Theorem 4.15 page 21} $$

<figure><img src="image_32.png" /></figure>

**Theorem 3.18** (Cantor intersection theorem) <sup>30</sup> Let (X, d) be a DISTANCE SPACE (Definition 3.1 page 6), $((A_n)_{n \in \mathbb{Z}})$ a SEQUENCE with each $A_n \in 2^X$, and $|A|$ the number of elements in A.

$$ \left\{ \begin{array}{ll} 1. & (X, d) \text{ is complete} \\ 2. & A_n \text{ is closed} \\ 3. & \text{diam } A_n \ge \text{diam } A_{n+1} \\ 4. & \text{diam } (A_n)_{n \in \mathbb{Z}} \to 0 \end{array} \right. \quad \left. \begin{array}{ll} \text{(Definition 3.13 page 9)} & \text{and} \\ \text{(Definition A.1 page 26)} & \text{and} \\ \text{(Definition 3.2 page 6)} & \text{and} \\ \text{(Definition 3.11 page 9)} & \text{and} \end{array} \right\} \implies \left\{ \left| \bigcap_{n \in \mathbb{N}} A_n \right| = 1 \right\} $$

<figure><img src="image_33.png" /></figure>

PROOF:

(1) Proof that $|\bigcap_{n \in \mathbb{Z}} A_n| < 2$:

(a) Let $A \triangleq \bigcap A_n$.

(b) $x \ne y$ and $\{x, y\} \in A \implies d(x, y) > 0$ and $\{x, y\} \subseteq A_n \forall n$

(c) $\exists n$ such that $\text{diam } A_n < d(x, y)$ by left hypothesis 4

(d) $\implies \exists n$ such that $\sup \{d(x, y) \mid x, y \in A_n\} < d(x, y)$

(e) This is a contradiction, so $\{x, y\} \notin A$ and $|\bigcap A_n| < 2$.

(2) Proof that $|\bigcap A_n| \ge 1$:

(a) Let $x_n \in A_n$ and $x_m \in A_m$

(b) $\forall \varepsilon, \exists N \in \mathbb{N}$ such that $A_N < \varepsilon$

(c) $\forall m, n > N, x_n \in A_n \subseteq A_N$ and $x_m \in A_m \subseteq A_N$

(d) $d(x_n, x_m) \le \text{diam } A_N < \varepsilon \implies \{x_n\}$ is a Cauchy sequence

(e) Because $\{x_n\}$ is complete, $x_n \to x$.

(f) $\implies x \in (A_n)^- = A_n$

(g) $\implies |A_n| \ge 1$

<figure><img src="image_34.png" /></figure>

**Definition 3.19** <sup>31</sup> Let $(X, d)$ be a distance space. Let $C$ be the set of all convergent sequences in $(X, d)$. The distance function $d$ is **continuous** in $(X, d)$ if

$$ ((x_n), (y_n) \in C \implies \lim_{n \to \infty} (d(x_n, y_n)) = d(\lim_{n \to \infty} (x_n), \lim_{n \to \infty} (y_n)). $$

A distance function is **discontinuous** if it is not continuous.

<figure><img src="image_35.png" /></figure>

<figure><img src="image_36.png" /></figure>

<footnote n="30"><sup>30</sup> in metric space: [Davis(2005)], page 28, [Hausdorff(1937)], page 150</footnote>

<footnote n="31"><sup>31</sup> [Blumenthal(1953)] page 9 (DEFINITION 6.3)</footnote>
**Remark 3.20** Rather than defining continuity of a distance function in terms of the sequential characterization of continuity as in Definition 3.19 (previous), we could define continuity using an inverse image characterization of continuity" (Definition 3.9 page 8). Assuming an equivalent topological space is used for both characterizations, the two characterizations are equivalent (Theorem A.20 page 29). In fact, one could construct an equivalence such as the following:

$$ \left\{ \begin{array}{l} \text{d is continuous in } \mathbb{R}^{X^2} \\ \text{(Definition A.11 page 28)} \\ \text{(inverse image characterization of continuity)} \end{array} \right\} \iff \left\{ \begin{array}{l} \lim_{(x_n) \in C} \left( \left( \text{d}(x_n, y_n) \right) \right) = \text{d} \left( \lim_{n \to \infty} (x_n), \lim_{n \to \infty} (y_n) \right) \\ \text{(Definition A.16 page 28)} \\ \text{(sequential characterization of continuity)} \end{array} \right\} $$

Note that just as $(x_n)$ is a sequence in $X$, so the ordered pair $((x_n), (y_n))$ is a sequence in $X^2$. The remainder follows from Theorem A.20 (page 29). However, use of the inverse image characterization is somewhat troublesome because we would need a topology on $X^2$, and we don't immediately have one defined and ready to use. In fact, we don't even immediately have a distance space on $X^2$ defined or even open balls in such a distance space. The result is, for the scope of this paper, it is arguably not worthwhile constructing the extra structure, but rather instead this paper uses the sequential characterization as a definition (as in Definition 3.19).

## 3.4 Examples

Similar distance functions and several of the observations for the examples in this section can be found in [Blumenthal(1953)] pages 8–13.

In a *metric space*, all *open balls* are *open*, the *open balls* form a *base* for a *topology*, the limits of *convergent sequences* are *unique*, and the *metric function* is *continuous*. In the *distance space* of the next example, none of these properties hold.

**Example 3.21** <fnref n="32" /> Let $(x, y)$ be an *ordered pair* in $\mathbb{R}^2$. Let $(a : b)$ be an *open interval* and $(a : b]$ a *half-open interval* in $\mathbb{R}$. Let $|x|$ be the *absolute value* of $x \in \mathbb{R}$. The function $d(x, y) \in \mathbb{R}^{\mathbb{R} \times \mathbb{R}}$ such that

$$ d(x, y) \triangleq \left\{ \begin{array}{ll} y & \forall (x, y) \in \{4\} \times (0 : 2] \quad \text{(vertical half-open interval)} \\ x & \forall (x, y) \in (0 : 2] \times \{4\} \quad \text{(horizontal half-open interval)} \\ |x - y| & \text{otherwise} \end{array} \right\} \quad \text{is a distance on } \mathbb{R}. $$

Note some characteristics of the *distance space* $(\mathbb{R}, d)$:

(1) $(\mathbb{R}, d)$ is not a *metric space* because $d$ does not satisfy the *triangle inequality*:
$$ d(0, 4) \triangleq |0 - 4| = 4 \not\le 2 = |0 - 1| + 1 \triangleq d(0, 1) + d(1, 4) $$

(2) Not every *open ball* in $(\mathbb{R}, d)$ is *open*. For example, the *open ball* $B(3, 2)$ is *not open* because $4 \in B(3, 2)$ but for all $0 < \varepsilon < 1$
$$ B(4, \varepsilon) = (4 - \varepsilon : 4 + \varepsilon) \cup (0 : \varepsilon) \not\subseteq (1 : 5) = B(3, 2) $$

(3) The *open balls* of $(\mathbb{R}, d)$ do not form a *base* for a *topology* on $\mathbb{R}$.
This follows directly from item (2) and Theorem 3.10 (page 8).

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

<footnote n="32">³² A similar distance function $d$ and item (4) page 13 can in essence be found in [Blumenthal(1953)] page 8. Definitions for Example 3.21: $(x, y)$: Definition 2.3 (page 4); $(a : b)$ and $(a : b]$: Definition 2.9 (page 5); $|x|$: Definition 2.10 (page 5); $\mathbb{R}^{\mathbb{R} \times \mathbb{R}}$: Definition 2.3 (page 4); *distance*: Definition 3.1 (page 6); *open ball*: Definition 3.5 (page 6); *open*: Definition 3.6 (page 7); *base*: Definition A.3 (page 27); *topology*: Definition A.1 (page 26); *open set*: Definition 3.6 (page 7); *topological space induced by* $(\mathbb{R}, d)$: Definition 3.9 (page 8); *discontinuous*: Definition 3.19 (page 11);</footnote>
(4) In the distance space $(\mathbb{R}, d)$, limits are not unique;

For example, the sequence $(\frac{1}{n})_1^\infty$ converges both to the limit 0 and the limit 4 in $(\mathbb{R}, d)$:

$$ \lim_{n \to \infty} d(x_n, 0) \triangleq \lim_{n \to \infty} d(\frac{1}{n}, 0) \triangleq \lim_{n \to \infty} |\frac{1}{n} - 0| = 0 \implies (\frac{1}{n}) \to 0 $$

$$ \lim_{n \to \infty} d(x_n, 4) \triangleq \lim_{n \to \infty} d(\frac{1}{n}, 4) \triangleq \lim_{n \to \infty} (\frac{1}{n}) = 0 \implies (\frac{1}{n}) \to 4 $$

(5) The topological space $(X, T)$ induced by $(\mathbb{R}, d)$ also yields limits of 0 and 4 for the sequence $(\frac{1}{n})_1^\infty$, just as it does in item (4). This is largely due to the fact that, for small $\epsilon$, the open balls $B(0, \epsilon)$ and $B(4, \epsilon)$ are open.

$$ \begin{array}{l} \text{B}(0, \epsilon) \text{ is open} \implies \text{for each } U \in T \text{ that contains } 0, \exists N \in \mathbb{N} \text{ such that } \frac{1}{n} \in U \quad \forall n > N \\ \qquad \Leftrightarrow (\frac{1}{n}) \to 0 \quad \text{by definition of convergence (Definition A.16 page 28)} \end{array} $$

$$ \begin{array}{l} \text{B}(4, \epsilon) \text{ is open} \implies \text{for each } U \in T \text{ that contains } 4, \exists N \in \mathbb{N} \text{ such that } \frac{1}{n} \in U \quad \forall n > N \\ \qquad \Leftrightarrow (\frac{1}{n}) \to 4 \quad \text{by definition of convergence (Definition A.16 page 28)} \end{array} $$

(6) The distance function $d$ is discontinuous (Definition 3.19 page 11):

$$ \begin{align*} \lim_{n \to \infty} (d(1 - \frac{1}{n}, 4 - \frac{1}{n})) &= \lim_{n \to \infty} (|(1 - \frac{1}{n}) - (4 - \frac{1}{n})|) = |1 - 4| = 3 \neq 4 = d(0, 4) \\ &= d\left(\lim_{n \to \infty} (1 - \frac{1}{n}), \lim_{n \to \infty} (4 - \frac{1}{n})\right) \end{align*} $$

In a *metric space*, all convergent sequences are also *Cauchy*. However, this is not the case for all *distance spaces*, as demonstrated next:

**Example 3.22** $^{33}$ The function $d(x, y) \in \mathbb{R}^{\mathbb{R} \times \mathbb{R}}$ such that

$$ d(x, y) \triangleq \left\{ \begin{array}{ll} |x - y| & \text{for } x = 0 \text{ or } y = 0 \text{ or } x = y \quad (\text{Euclidean}) \\ 1 & \text{otherwise} \end{array} \right\} \quad \text{is a distance on } \mathbb{R}. $$

Note some characteristics of the distance space $(\mathbb{R}, d)$:

(1) $(X, d)$ is not a *metric space* because the *triangle inequality* does not hold:

$$ d(\frac{1}{4}, \frac{1}{2}) = 1 \le \frac{3}{4} = |\frac{1}{4} - 0| + |0 - \frac{1}{2}| = d(\frac{1}{4}, 0) + d(0, \frac{1}{2}) $$

(2) The *open ball* $B(\frac{1}{4}, \frac{1}{2})$ is *not open* because for any $\epsilon \in \mathbb{R}^+$, no matter how small,

$$ B(0, \epsilon) = (-\epsilon : +\epsilon) \not\subseteq \{0, \frac{1}{4}\} = \{x \in X \mid d(\frac{1}{4}, x) < \frac{1}{2}\} \triangleq B(\frac{1}{4}, \frac{1}{2}) $$

(3) Even though not all the *open balls* are *open*, it is still possible to have an *open set* in $(X, d)$. For example, the set $U \triangleq \{1, 2\}$ is *open*:

$$ B(1, 1) \triangleq \{x \in X \mid d(1, x) < 1\} = \{1\} \subseteq \{1, 2\} \triangleq U $$

$$ B(2, 1) \triangleq \{x \in X \mid d(2, x) < 1\} = \{2\} \subseteq \{1, 2\} \triangleq U $$

(4) By item (2) and Theorem 3.10 (page 8), the *open balls* of $(\mathbb{R}, d)$ do not form a *base* for a *topology* on $\mathbb{R}$.

(5) Even though the open balls in $(\mathbb{R}, d)$ do not induce a topology on $X$, it is still possible to find a set of *open sets* in $(X, d)$ that is a topology. For example, the set $\{\emptyset, \{1, 2\}, \mathbb{R}\}$ is a topology on $\mathbb{R}$.

(6) In $(\mathbb{R}, d)$, limits of convergent sequences are unique:

$$ (x_n) \to x \implies \lim_{n \to \infty} d(x_n, x) = \left\{ \begin{array}{ll} \lim_{n \to \infty} |x_n - 0| & = 0 & \text{for } x = 0 \\ |x - x| & = 0 & \text{for constant } (x_n) \text{ for } n > N \\ 1 & \neq 0 & \text{otherwise} \end{array} \right. \quad \begin{array}{l} \text{OR} \\ \text{OR} \end{array} $$

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

<footnote n="33">33 The distance function $d$ and item (7) page 14 can in essence be found in [Blumenthal (1953)] page 9</footnote>
which says that there are only two ways for a sequence to converge: either $x = 0$ or the sequence eventually becomes constant (or both). Any other sequence will diverge. Therefore we can say the following:

(a) If $x = 0$ and the sequence is not constant, then the limit is *unique* and 0.

(b) If $x = 0$ and the sequence is constant, then the limit is *unique* and 0.

(c) If $x \neq 0$ and the sequence is constant, then the limit is *unique* and $x$.

(d) If $x \neq 0$ and the sequence is not constant, then the sequence diverges and there is no limit.

(7) In $(\mathbb{R}, \text{d})$, a convergent sequence is not necessarily *Cauchy*. For example,

(a) the sequence $(1/n)_{n \in \mathbb{N}}$ is convergent with limit $0$: $\lim_{n \to \infty} d(1/n, 0) = \lim_{n \to \infty} 1/n = 0$

(b) However, even though $(1/n)$ is convergent, it is *not Cauchy*: $\lim_{n,m \to \infty} d(1/n, 1/m) = 1 \neq 0$

(8) The distance function $d$ is discontinuous in $(X, \text{d})$:

$$ \lim_{n \to \infty} (d(1/n, 2 - 1/n)) = 1 \neq 2 = d(0, 2) = d\left(\lim_{n \to \infty} (1/n), \lim_{n \to \infty} (2 - 1/n)\right). $$

**Example 3.23** $^{34}$ The function $d(x, y) \in \mathbb{R}^{\mathbb{R} \times \mathbb{R}}$ such that

$$ d(x, y) \triangleq \begin{cases} 2|x - y| & \forall (x, y) \in \{(0, 1), (1, 0)\} \quad \text{(dilated Euclidean)} \\ |x - y| & \text{otherwise} \end{cases} \quad \text{is a distance on } \mathbb{R}. $$

Note some characteristics of the *distance space* $(\mathbb{R}, \text{d})$:

(1) $(\mathbb{R}, \text{d})$ is *not a metric space* because $d$ does *not* satisfy the *triangle inequality*:

$$ d(0, 1) \triangleq 2|0 - 1| = 2 \not\le 1 = |0 - \frac{1}{2}| + |\frac{1}{2} - 1| \triangleq d(0, \frac{1}{2}) + d(\frac{1}{2}, 1) $$

(2) The function $d$ is discontinuous:

$$ \lim_{n \to \infty} (d(1 - 1/n, 1/n)) \triangleq \lim_{n \to \infty} (|1 - 1/n - 1/n|) = 1 \neq 2 = 2|0 - 1| \triangleq d(0, 1) = d\left(\lim_{n \to \infty} (1 - 1/n), \lim_{n \to \infty} (1/n)\right). $$

(3) In $(X, \text{d})$, *open balls* are *open*:

(a) $p(x, y) \triangleq |x - y|$ is a *metric* and thus all open balls in that do not contain both 0 and 1 are *open*.

(b) By Example C.4 (page 36), $q(x, y) \triangleq 2|x - y|$ is also a *metric* and thus all open balls containing 0 and 1 only are *open*.

(c) The only question remaining is with regards to open balls that contain 0, 1 and some other element(s) in $\mathbb{R}$. But even in this case, open balls are still open. For example:

$$ B(-1, 2) = (-1 : 2) = (-1 : 1) \cup (1 : 2) $$

Note that both $(-1 : 1)$ and $(1 : 2)$ are *open*, and thus by Theorem 3.7 (page 7), $B(-1, 2)$ is *open* as well.

(4) By item (3) and Theorem 3.10 (page 8), the *open balls* of $(\mathbb{R}, \text{d})$ do form a *base* for a *topology* on $\mathbb{R}$.

(5) In $(X, \text{d})$, the limits of convergent sequences are *unique*. This is demonstrated in Example 4.22 (page 25) using additional structure developed in Section 4.

(6) In $(X, \text{d})$, convergent sequences are *Cauchy*. This is also demonstrated in Example 4.22 (page 25).

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

<footnote n="34">34 The distance function $d$ and item (2) page 14 can in essence be found in [Blumenthal (1953)] page 9</footnote>
The distance functions in Example 3.21 (page 12)–Example 3.23 (page 14) were all discontinuous. In the absence of the triangle inequality and in light of these examples, one might try replacing the triangle inequality with the weaker requirement of continuity. However, as demonstrated by the next example, this also leads to an arguably disastrous result.

**Example 3.24**<sup>35</sup> The function $d \in \mathbb{R}^{\mathbb{R} \times \mathbb{R}}$ such that $d(x, y) \triangleq (x - y)^2$ is a distance on $\mathbb{R}$.
Note some characteristics of the distance space $(\mathbb{R}, d)$:

(1) $(\mathbb{R}, d)$ is not a metric space because the triangle inequality does not hold:
$$d(0, 2) \triangleq (0 - 2)^2 = 4 \le 2 = (0 - 1)^2 + (1 - 2)^2 \triangleq d(0, 1) + d(1, 2)$$

(2) The distance function $d$ is continuous in $(X, d)$. This is demonstrated in the more general setting of Section 4 in Example 4.23 (page 25).

(3) Calculating the length of curves in $(X, d)$ leads to a paradox:<fnref n="36" />

(a) Partition $[0 : 1]$ into $2^N$ consecutive line segments connected at the points
$$((0, \frac{1}{2^N}, \frac{2}{2^N}, \frac{3}{2^N}, \dots, \frac{2^{N-1}}{2^N}, 1))$$

(b) Then the distance, as measured by $d$, between any two consecutive points is
$$d(p_n, p_{n+1}) \triangleq (p_n - p_{n+1})^2 = (\frac{1}{2^N})^2 = \frac{1}{2^{2N}}$$

(c) But this leads to the paradox that the total length of $[0 : 1]$ is 0:

$$\lim_{N \to \infty} \sum_{n=0}^{2^N-1} \frac{1}{2^{2N}} = \lim_{N \to \infty} \frac{2^N}{2^{2N}} = \lim_{N \to \infty} \frac{1}{2^N} = 0$$

# 4 Distance spaces with power triangle inequalities

## 4.1 Definitions

This paper introduces a new relation called the power triangle inequality (Definition 4.3 page 16). It is a generalization of other common relations, including the triangle inequality (Definition 4.4 page 16). The power triangle inequality is defined in terms of a function herein called the power triangle function (next definition). This function is a special case of the power mean with $N = 2$ and $\lambda_1 = \lambda_2 = \frac{1}{2}$ (Definition B.6 page 31). Power means have the attractive properties of being continuous and strictly monontone with respect to a free parameter $p \in \mathbb{R}^*$ (Theorem B.7 page 31). This fact is inherited and exploited by the power triangle inequality (Corollary 4.6 page 16).

**Definition 4.1** Let $(X, d)$ be a distance space (Definition 3.1 page 6). Let $\mathbb{R}^+$ be the set of all positive real numbers and $\mathbb{R}^*$ be the set of extended real numbers (Definition 2.1 page 4). The power triangle function $\tau$ on $(X, d)$ is defined as

$$\tau(p, \sigma; x, y, z; d) \triangleq 2\sigma\left[\frac{1}{2}d^p(x, z) + \frac{1}{2}d^p(z, y)\right]^{\frac{1}{p}} \quad \forall (p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+; x, y, z \in X$$

<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /></figure>

<footnote n="35">35 [Blumenthal (1953)] pages 12–13, [Laos (1998)] pages 118–119</footnote>

<footnote n="36">36 This is the method of “inscribed polygons” for calculating the length of a curve and goes back to Archimedes: [Bunschwig et al. (2003) Bunschwig, Lloyd, and Pellegrin] page 26, [Walmsley (1920)], page 200 (§158),</footnote>
**Remark 4.2**<sup>37</sup> In the field of probabilistic metric spaces, a function called the *triangle function* was introduced by Sherstnev in 1962. However, the *power triangle function* as defined in this present paper is *not* a special case of (is not compatible with) the *triangle function* of Sherstnev. Another definition of *triangle function* has been offered by Bessenyei in 2014 with special cases of $\Phi(u, v) \triangleq c(u + v)$ and $\Phi(u, v) \triangleq (u^p + v^p)^{\frac{1}{p}}$, which *are* similar to the definition of *power triangle function* offered in this present paper.

**Definition 4.3** Let $(X, d)$ be a *distance space*. Let $2^{XXX}$ be the set of all trinomial *relations* (Definition 2.3 page 4) on $X$. A relation $\oplus(p, \sigma; d)$ in $2^{XXX}$ is a **power triangle inequality** on $(X, d)$ if

$$ \oplus(p, \sigma; d) \triangleq \{(x, y, z) \in \mathbb{X}^3 \mid d(x, y) \le \tau(p, \sigma; x, y, z; d)\} \quad \text{for some } (p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+. $$

The tuple $(X, d, p, \sigma)$ is a **power distance space** and $d$ a **power distance or power distance function** if $(X, d)$ is a *distance space* in which the *triangle relation* $\oplus(p, \sigma; d)$ holds.

The *power triangle function* can be used to define some standard inequalities (next definition). See *Corollary 4.7* (page 17) for some justification of the definitions.

**Definition 4.4**<sup>38</sup> Let $\oplus(p, \sigma; d)$ be a *power triangle inequality* on a *distance space* $(X, d)$.

1. $\oplus(\infty, \sigma/2; d)$ is the **$\sigma$-inframetric inequality**

2. $\oplus(\infty, \frac{1}{2}; d)$ is the **inframetric inequality**

3. $\oplus(2, \sqrt{2}/2; d)$ is the **quadratic inequality**

4. $\oplus(1, \sigma; d)$ is the **relaxed triangle inequality**

5. $\oplus(1, 1; d)$ is the **triangle inequality**

6. $\oplus(1/2, 2; d)$ is the **square mean root inequality**

7. $\oplus(0, 1/2; d)$ is the **geometric inequality**

8. $\oplus(-1, 1/4; d)$ is the **harmonic inequality**

9. $\oplus(-\infty, 1/2; d)$ is the **minimal inequality**

**Definition 4.5**<sup>39</sup> Let $(X, d)$ be a *distance space* (Definition 3.1 page 6).

1. $(X, d)$ is a **metric space** if the *triangle inequality* holds in $X$.

2. $(X, d)$ is a **near metric space** if the *relaxed triangle inequality* holds in $X$.

3. $(X, d)$ is an **inframetric space** if the *inframetric inequality* holds in $X$.

4. $(X, d)$ is a **$\sigma$-inframetric space** if the *$\sigma$-inframetric inequality* holds in $X$.

## 4.2 Properties

### 4.2.1 Relationships of the power triangle function

**Corollary 4.6** Let $\tau(p, \sigma; x, y, z; d)$ be the POWER TRIANGLE FUNCTION (Definition 4.1 page 15) in the DISTANCE SPACE (Definition 3.1 page 6) $(X, d)$. Let $(\mathbb{R}, |\cdot|, \le)$ be the ORDERED METRIC SPACE with the usual ordering relation

<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /></figure>

<footnote n="37"><sup>37</sup> ⇄ [Sherstnev(1962)], page 4, ⇄ [Schweizer and Sklar(1983)] page 9 (1.6.1)-(1.6.4), ⌜ [Bessenyei and Pales(2014)] page 2</footnote>

<footnote n="38"><sup>38</sup> ⌜ [Bessenyei and Pales(2014)] page 2, ⌜ [Czerwik(1993)] page 5 (b-metric; (1),(2),(5)), ⌜ [Fagin et al.(2003a)Fagin, Kumar, and Sivakumar], ⌜ [Fagin et al.(2003b)Fagin, Kumar, and Sivakumar] (Definition 4.2 (Relaxed metrics)), ⌜ [Xia(2009)] page 453 (Definition 2.1), ⇄ [Heinonen(2001)] page 109 (14.1 Quasimetric spaces.), ⇄ [Kirk and Shahzad(2014)] page 113 (Definition 12.1), ⇄ [Deza and Deza(2014)] page 7, ⌜ [Hoehn and Niven(1985)] page 151, ⇄ [Gibbons et al.(1977)Gibbons, Olkin, and Sobel] page 51 (square-mean-root (SMR) (2.4.1)), ⇄ [Euclid(circa 300BC)] (triangle inequality—Book I Proposition 20)</footnote>

<footnote n="39"><sup>39</sup> **metric space**: ⇄ [Dieudonné(1969)], page 28, ⇄ [Copson(1968)], page 21, ⇄ [Hausdorff(1937)] page 109, ⇄ [Fréchet(1928)], ⌜ [Fréchet(1906)] page 30 **near metric space**: ⌜ [Czerwik(1993)] page 5 (b-metric; (1),(2),(5)), ⌜ [Fagin et al.(2003a)Fagin, Kumar, and Sivakumar], ⌜ [Fagin et al.(2003b)Fagin, Kumar, and Sivakumar] (Definition 4.2 (Relaxed metrics)), ⌜ [Xia(2009)] page 453 (Definition 2.1), ⇄ [Heinonen(2001)] page 109 (14.1 Quasimetric spaces.), ⇄ [Kirk and Shahzad(2014)] page 113 (Definition 12.1), ⇄ [Deza and Deza(2014)] page 7</footnote>
≤ and usual metric $|\cdot|$ on $\mathbb{R}$. The function $\tau(p, \sigma; x, y, z; d)$ is CONTINUOUS and STRICTLY MONOTONE in $(\mathbb{R}, |\cdot|, \le)$ with respect to both the variables $p$ and $\sigma$.

<figure><img src="image_47.png" /></figure>

**PROOF:**

(1) Proof that $\tau(p, \sigma; x, y, z; d)$ is continuous and strictly monotone with respect to $p$: This follows directly from Theorem B.7 (page 31).

(2) Proof that $\tau(p, \sigma; x, y, z; d)$ is continuous and strictly monotone with respect to $\sigma$:

$$ 
\begin{align*}
\tau(p, \sigma; x, y, z; d) &\triangleq 2\sigma \underbrace{\left[ \frac{1}{2} d^p(x, z) + \frac{1}{2} d^p(z, y) \right]}_{f(p, x, y, z)}^{\frac{1}{p}} & \text{by definition of } \tau \text{ (Definition 4.1 page 15)} \\
&= 2\sigma f(p, x, y, z) & \text{where } f \text{ is defined as above} \\
&\Rightarrow \tau \text{ is affine with respect to } \sigma \\
&\Rightarrow \tau \text{ is continuous and strictly monotone with respect to } \sigma:
\end{align*}
$$

<figure><img src="image_48.png" /></figure>

**Corollary 4.7** Let $\tau(p, \sigma; x, y, z; d)$ be the POWER TRIANGLE FUNCTION in the DISTANCE SPACE (Definition 3.1 page 6) $(X, d)$.

$$ \tau(p, \sigma; x, y, z; d) = \begin{cases} 2\sigma \max \{d(x, z), d(z, y)\} & \text{for } p = \infty, \quad (\text{MAXIMUM, corresponds to INFRAMETRIC SPACE}) \\ 2\sigma \left[ \frac{1}{2} d^2(x, z) + \frac{1}{2} d^2(z, y) \right]^{\frac{1}{2}} & \text{for } p = 2, \quad (\text{QUADRATIC MEAN}) \\ \sigma[d(x, z) + d(z, y)] & \text{for } p = 1, \quad (\text{ARITHMETIC MEAN, corresponds to NEAR METRIC SPACE}) \\ 2\sigma \sqrt{d(x, z)} \sqrt{d(z, y)} & \text{for } p = 0, \quad (\text{GEOMETRIC MEAN}) \\ 4\sigma \left[ \frac{1}{d(x,z)} + \frac{1}{d(z,y)} \right]^{-1} & \text{for } p = -1, \quad (\text{HARMONIC MEAN}) \\ 2\sigma \min \{d(x, z), d(z, y)\} & \text{for } p = -\infty, \quad (\text{MINIMUM}) \end{cases} $$

<figure><img src="image_49.png" /></figure>

**PROOF:** These follow directly from Theorem B.7 (page 31).

<figure><img src="image_50.png" /></figure>

**Corollary 4.8** Let $(X, d)$ be a DISTANCE SPACE.

$$ 
\begin{align*}
2\sigma \min \{d(x, z), d(z, y)\} &\le 4\sigma \left[ \frac{1}{d(x,z)} + \frac{1}{d(z,y)} \right]^{-1} \le 2\sigma \sqrt{d(x, z)} \sqrt{d(z, y)} \\
&\le \sigma[d(x, z) + d(z, y)] \le 2\sigma \max \{d(x, z), d(z, y)\}
\end{align*}
$$

<figure><img src="image_51.png" /></figure>

**PROOF:** These follow directly from Corollary B.8 (page 34).

<figure><img src="image_52.png" /></figure>

### 4.2.2 Properties of power distance spaces

The power triangle inequality property of a power distance space axiomatically endows a metric with an upper bound. Lemma 4.9 (next) demonstrates that there is a complementary lower bound somewhat similar in form to the power triangle inequality upper bound. In the special case where $2\sigma = 2^{\frac{1}{p}}$, the lower bound helps provide a simple proof of the continuity of a large class of power distance functions (Theorem 4.18 page 23). The inequality $2\sigma \le 2^{\frac{1}{p}}$ is a special relation in this paper and appears repeatedly in this paper; it appears as an inequality in Lemma 4.13 (page 20), Corollary 4.12 (page 20) and Corollary 4.14 (page 21), and as an equality in Lemma 4.9 (next) and Theorem 4.18 (page 23). It is plotted in Figure 1 (page 18).

<figure><img src="image_53.png" /></figure>

<figure><img src="image_54.png" /></figure>
<figure><img src="image_55.png" /><figcaption>Figure 1: $\sigma = \frac{1}{2}(2^{\frac{1}{p}}) = 2^{\frac{1}{p}-1}$ or $p = \frac{\ln 2}{\ln(2\sigma)}$ (see Lemma 4.9 page 18, Lemma 4.13 page 20, Corollary 4.14 page 21, Corollary 4.12 page 20, and Theorem 4.18 page 23).</figcaption></figure>

**Lemma 4.9**<sup>40</sup> Let $(X, d, p, \sigma)$ be a POWER TRIANGLE TRIANGLE SPACE (Definition 4.3 page 16). Let $|\cdot|$ be the ABSOLUTE VALUE function (Definition 2.10 page 5). Let $\max\{x, y\}$ be the maximum and $\min\{x, y\}$ the minimum of any $x, y \in \mathbb{R}^*$. Then, for all $(p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+$,

$$1. \quad d^p(x, y) \ge \max \left\{ 0, \frac{2}{(2\sigma)^p} d^p(x, z) - d^p(z, y), \frac{2}{(2\sigma)^p} d^p(y, z) - d^p(z, x) \right\} \quad \forall x, y, z \in X \quad \text{and}$$

$$2. \quad d(x, y) \ge |d(x, z) - d(z, y)| \quad \text{if } p \ne 0 \text{ and } 2\sigma = 2^{\frac{1}{p}} \quad \forall x, y, z \in X.$$

<figure><img src="image_56.png" /></figure>

PROOF:

$$\text{(1) lemma: } \frac{2}{(2\sigma)^p} d^p(x, z) - d^p(z, y) \le d^p(x, y) \quad \forall (p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+: \text{ Proof:}$$

$$\begin{align*}
\frac{2}{(2\sigma)^p} d^p(x, z) - d^p(z, y) &\le \frac{2}{(2\sigma)^p} \left[ 2\sigma \left[ \frac{1}{2} d^p(x, y) + \frac{1}{2} d^p(y, z) \right]^{\frac{1}{p}} \right]^p - d^p(z, y) && \text{by power triangle inequality} \\
&= \frac{2(2\sigma)^p}{(2\sigma)^p} \left[ \frac{1}{2} d^p(x, y) + \frac{1}{2} d^p(y, z) \right] - d^p(z, y) \\
&= [d^p(x, y) + d^p(y, z)] - d^p(y, z) && \text{by symmetric property of d} \\
&= d^p(x, y)
\end{align*}$$

$$\text{(2) Proof for } (p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+ \text{ case:}$$

$$d^p(x, y) \ge \frac{2}{(2\sigma)^p} d^p(x, z) - d^p(z, y) \quad \text{by (1) lemma}$$

$$d^p(x, y) = d^p(y, x) \ge \frac{2}{(2\sigma)^p} d^p(y, z) - d^p(z, x) \quad \text{by commutative property of d and (1) lemma}$$

$$d^p(x, y) \ge 0 \quad \text{by non-negative property of d (Definition 3.1 page 6)}$$

The rest follows because $g(x) \triangleq x^{\frac{1}{p}}$ is strictly monotone in $\mathbb{R}^\mathbb{R}$.

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /></figure>

<footnote n="40"><sup>40</sup> in metric space ((p, σ) = (1, 1)): [Dieudonné(1969)] page 28, [Michel and Herget(1993)] page 266,</footnote>

<footnote>< [Berberian(1961)] page 37 (Theorem II.4.1)</footnote>
(3) Proof for $2\sigma = 2^{\frac{1}{p}}$ case:

$$ 
\begin{align*}
d(x, y) &\ge \max \left\{ 0, \frac{2}{(2\sigma)^p} d^p(x, z) - d^p(z, y), \frac{2}{(2\sigma)^p} d^p(y, z) - d^p(z, x) \right\}^{\frac{1}{p}} && \text{by item (2) (page 18)} \\
&= \max \{0, d(x, z) - d(z, y), d(y, z) - d(z, x)\} && \text{by } 2\sigma = 2^{\frac{1}{p}} \text{ hypothesis } \iff \frac{2}{(2\sigma)^p} = 1 \\
&= \max \{0, (d(x, z) - d(z, y)), -(d(x, z) - d(z, y))\} && \text{by symmetric property of d} \\
&= |(d(x, z) - d(z, y))| && \text{by symmetric property of d}
\end{align*}
$$

<figure><img src="image_59.png" /></figure>

**Theorem 4.10** Let $(X, d, p, \sigma)$ be a POWER DISTANCE SPACE (Definition 4.3 page 16). Let $B$ be an OPEN BALL (Definition 3.5 page 6) on $(X, d)$. Then for all $(p, \sigma) \in (\mathbb{R}^*\setminus\{0\}) \times \mathbb{R}^+$,

$$ 
\left\{ 
\begin{array}{ll}
\text{A.} & 2\sigma \le 2^{\frac{1}{p}} \\
\text{B.} & q \in \mathbf{B}(\theta, r)
\end{array}
\right\} \quad \text{and} \quad 
\left\{ 
\begin{array}{l}
\text{1.} \quad \exists r_q \in \mathbb{R}^+ \text{ such that} \\
\qquad \mathbf{B}(q, r_q) \subseteq \mathbf{B}(\theta, r)
\end{array}
\right\} \quad \implies \quad 
\left\{ 
\begin{array}{l}
\text{B.} \\
q \in \mathbf{B}(\theta, r)
\end{array}
\right\}
$$

<figure><img src="image_60.png" /></figure>

PROOF:

(1) lemma:

$$ 
\begin{align*}
q \in \mathbf{B}(\theta, r) &\iff d(\theta, q) < r && \text{by definition of open ball (Definition 3.5 page 6)} \\
&\iff 0 < r - d(\theta, q) && \text{by field property of real numbers} \\
&\iff \exists r_q \in \mathbb{R}^+ \text{ such that } 0 < r_q < r - d(\theta, q) && \text{by The Archimedean Property}^{41}
\end{align*}
$$

(2) Proof that (A), (B) $\implies$ (1):

$$ 
\begin{align*}
\mathbf{B}(q, r_q) &\triangleq \{x \in X \mid d(q, x) < r_q\} && \text{by definition of open ball (Definition 3.5 page 6)} \\
&= \{x \in X \mid d^p(q, x) < r_q^p \in \mathbb{R}^+\} && \text{because } f(x) \triangleq x^p \text{ is monotone} \\
&\subseteq \{x \in X \mid d^p(q, x) < r^p - d^p(\theta, q)\} && \text{by hypothesis B and (1) lemma page 19} \\
&= \{x \in X \mid d^p(\theta, q) + d^p(q, x) < r^p\} && \text{by field property of real numbers} \\
&= \{x \in X \mid [d^p(\theta, q) + d^p(q, x)]^{\frac{1}{p}} < r\} && \text{because } f(x) \triangleq x^{\frac{1}{p}} \text{ is monotone} \\
&\subseteq \{x \in X \mid 2^{1-\frac{1}{p}}\sigma [d^p(\theta, q) + d^p(q, x)]^{\frac{1}{p}} < r\} && \text{by hypothesis A which implies } 2^{1-\frac{1}{p}}\sigma \le 1 \\
&= \{x \in X \mid 2\sigma[\frac{1}{2}d(\theta, q)x + \frac{1}{2}d^p(q, x)]^{\frac{1}{p}} < r\} && \text{because } 2^{1-\frac{1}{p}}\sigma = 2\sigma(\frac{1}{2})^{\frac{1}{p}} \\
&\triangleq \{x \in X \mid \tau(p, \sigma, \theta, x, q) < r\} && \text{by definition of } \tau \text{ (Definition 4.1 page 15)} \\
&\subseteq \{x \in X \mid d(\theta, x) < r\} && \text{by definition of } (X, d, p, \sigma) \text{ (Definition 4.3 page 16)} \\
&\triangleq \mathbf{B}(\theta, r) && \text{by definition of open ball (Definition 3.5 page 6)}
\end{align*}
$$

(3) Proof that (B) $\Leftarrow$ (1):

$$ 
\begin{align*}
q \in \{x \in X \mid d(q, x) = 0\} && \text{by nondegenerate property (Definition 3.1 page 6)} \\
&\subseteq \{x \in X \mid d(q, x) < r_q\} && \text{because } r_q > 0 \\
&\triangleq \mathbf{B}(q, r_q) && \text{by definition of open ball (Definition 3.5 page 6)} \\
&\subseteq \mathbf{B}(\theta, r) && \text{by hypothesis 2}
\end{align*}
$$

<figure><img src="image_61.png" /></figure>

<figure><img src="image_62.png" /></figure>

<footnote n="41">41 [Aliprantis and Burkinshaw(1998)] page 17 (Theorem 3.3 ("The Archimedean Property") and Theorem 3.4), [Zorich(2004)] page 53 (6° ("The principle of Archimedes") and 7°)</footnote>
<figure><img src="image_63.png" /></figure>

**Corollary 4.11** Let $(X, d, p, \sigma)$ be a POWER DISTANCE SPACE. Then for all $(p, \sigma) \in (\mathbb{R}^* \setminus \{0\}) \times \mathbb{R}^+$,
$$ \{2\sigma \le 2^{\frac{1}{p}}\} \Rightarrow \{\text{every OPEN BALL in } (X, d) \text{ is OPEN}\} $$

PROOF: This follows from Theorem 4.10 (page 19) and Theorem 3.10 (page 8).

<figure><img src="image_64.png" /></figure>

**Corollary 4.12** Let $(X, d, p, \sigma)$ be a POWER DISTANCE SPACE. Let $B$ be the set of all OPEN BALLS in $(X, d)$. Then for all $(p, \sigma) \in (\mathbb{R}^* \setminus \{0\}) \times \mathbb{R}^+$,
$$ \{2\sigma \le 2^{\frac{1}{p}}\} \Rightarrow \{\text{B is a BASE for } (X, T)\} $$

PROOF:

(1) The set of all open balls in $(X, d)$ is a base for $(X, T)$ by Corollary 4.11 (page 20) and Theorem A.4 (page 27).

(2) $T$ is a topology on $X$ by Definition A.3 (page 27).

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /><figcaption>Figure 2: open set (see Lemma 4.13 page 20)</figcaption></figure>

Lemma 4.13 (next) demonstrates that every point in an open set is contained in an open ball that is contained in the original open set (see also Figure 2 page 20).

**Lemma 4.13** Let $(X, d, p, \sigma)$ be a POWER DISTANCE SPACE. Let $B$ be an OPEN BALL on $(X, d)$. Then for all $(p, \sigma) \in (\mathbb{R}^* \setminus \{0\}) \times \mathbb{R}^+$,
$$ \left\{ \begin{array}{l} A. \ 2\sigma \le 2^{\frac{1}{p}} \text{ and} \\ B. \ U \text{ is OPEN in } (X, d) \end{array} \right\} \Rightarrow \left\{ \begin{array}{l} 1. \ \forall x \in U, \exists r \in \mathbb{R}^+ \text{ such that} \\ \qquad B(x, r) \subseteq U \end{array} \right\} \Rightarrow \left\{ \begin{array}{l} B. \ U \text{ is} \\ \qquad \text{OPEN in } (X, d) \end{array} \right\} $$

PROOF:

(1) Proof that for $((A), (B) \Rightarrow (1))$:

$$ \begin{aligned} U &= \bigcup \{ \mathbf{B}(x_\gamma, r_\gamma) \mid \mathbf{B}(x_\gamma, r_\gamma) \subseteq U \} & \text{by left hypothesis and Corollary 4.12 page 20} \\ \supseteq \mathbf{B}(x, r) & \text{because x must be in one of those balls in } U \end{aligned} $$

<figure><img src="image_67.png" /></figure>

<figure><img src="image_68.png" /></figure>
<figure><img src="image_69.png" /></figure>

PROOF:

(1) Proof that convergent $\Rightarrow$ Cauchy:

$$
\begin{align*}
d(x_n, x_m) &\le \tau(p, \sigma; x_n, x_m, x) && \text{by definition of power triangle inequality (Definition 4.3 page 16)} \\
&\triangleq 2\sigma \left[ \frac{1}{2} d^p(x_n, x) + \frac{1}{2} d^p(x_m, x) \right]^{\frac{1}{p}} && \text{by definition of power triangle function (Definition 4.1 page 15)} \\
&< 2\sigma \left[ \frac{1}{2} \varepsilon^p + \frac{1}{2} \varepsilon^p \right]^{\frac{1}{p}} && \text{by convergence hypothesis (Definition A.16 page 28)} \\
&= 2\sigma\varepsilon && \text{by definition of convergence (Definition A.16 page 28)} \\
&\Rightarrow \text{Cauchy} && \text{by definition of Cauchy (Definition 3.12 page 9)}
\end{align*}
$$

$$
\begin{align*}
d(x_n, x_m) &\le \tau(\infty, \sigma; x_n, x_m, x) && \text{by definition of power triangle inequality at } p = \infty \\
&= 2\sigma \max \{d(x_n, x), d(x_m, x)\} && \text{by Corollary 4.7 (page 17)} \\
&= 2\sigma \max \{\varepsilon, \varepsilon\} && \text{by convergent hypothesis (Definition A.16 page 28)} \\
&= 2\sigma\varepsilon && \text{by definition of max}
\end{align*}
$$

$$
\begin{align*}
d(x_n, x_m) &\le \tau(-\infty, \sigma; x_n, x_m, x) && \text{by definition of power triangle inequality at } p = -\infty \\
&= 2\sigma \min \{d(x_n, x), d(x_m, x)\} && \text{by Corollary 4.7 (page 17)} \\
&= 2\sigma \min \{\varepsilon, \varepsilon\} && \text{by convergent hypothesis (Definition A.16 page 28)} \\
&= 2\sigma\varepsilon && \text{by definition of min}
\end{align*}
$$

$$
\begin{align*}
d(x_n, x_m) &\le \tau(0, \sigma; x_n, x_m, x) && \text{by definition of power triangle inequality at } p = 0 \\
&= 2\sigma \sqrt{d(x_n, x)} \sqrt{d(x_m, x)} && \text{by Corollary 4.7 (page 17)} \\
&= 2\sigma \sqrt{\varepsilon} \sqrt{\varepsilon} && \text{by convergent hypothesis (Definition A.16 page 28)} \\
&= 2\sigma\varepsilon && \text{by property of } \mathbb{R}
\end{align*}
$$

(2) Proof that Cauchy $\Rightarrow$ bounded: by Proposition 3.14 (page 9).

<figure><img src="image_70.png" /></figure>

**Theorem 4.17** 45 Let $(X, d, p, \sigma)$ be a POWER DISTANCE SPACE. Let $f \in \mathbb{Z}^\mathbb{Z}$ be a STRICTLY MONOTONE function such that $f(n) < f(n + 1)$. For any $(p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+$

$$
\left\{
\begin{array}{ll}
1. & ((x_n)_{n \in \mathbb{Z}} \text{ is CAUCHY and } ((x_{f(n)})_{n \in \mathbb{Z}} \text{ is CONVERGENT}) \\
2. &
\end{array}
\right.
\quad \Rightarrow \quad \{((x_n)_{n \in \mathbb{Z}} \text{ is CONVERGENT.})
$$

<figure><img src="image_71.png" /></figure>

PROOF:

$$
\begin{align*}
d(x_n, x) &= d(x, x_n) && \text{by symmetric property of d} \\
&\le \tau(p, \sigma; x, x_n, x_{f(n)}) && \text{by definition of power triangle inequality (Definition 4.3 page 16)} \\
&\triangleq 2\sigma \left[ \frac{1}{2} d^p(x, x_{f(n)}) + \frac{1}{2} d^p(x_{f(n)}, x_n) \right]^{\frac{1}{p}} && \text{by definition of power triangle function (Definition 4.1 page 15)} \\
&= 2\sigma \left[ \frac{1}{2} \varepsilon + \frac{1}{2} d^p(x_{f(n)}, x_n) \right]^{\frac{1}{p}} && \text{by left hypothesis 2} \\
&= 2\sigma \left[ \frac{1}{2} \varepsilon^p + \frac{1}{2} \varepsilon^p \right]^{\frac{1}{p}} && \text{by left hypothesis 1} \\
&= 2\sigma\varepsilon && \text{by definition of convergent (Definition A.16 page 28)} \\
&\Rightarrow \text{convergent} && \text{by definition of convergent (Definition A.16 page 28)}
\end{align*}
$$

<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /></figure>

<footnote n="45">⁴⁵ in *metric space*: [Rosenlicht(1968)] page 52</footnote>
<figure><img src="image_74.png" /></figure>

**Theorem 4.18** <sup>46</sup> Let $(X, d, p, \sigma)$ be a POWER DISTANCE SPACE. Let $(\mathbb{R}, q)$ be a metric space of real numbers with the usual metric $q(x, y) \triangleq |x - y|$. Then

$$ \{2\sigma = 2^{\frac{1}{p}}\} \implies \{ \text{d is continuous in } (\mathbb{R}, q) \} $$

<figure><img src="image_75.png" /></figure>

PROOF:

$$
\begin{align*}
|d(x, y) - d(x_n, y_n)| &\le |d(x, y) - d(x_n, y)| + |d(x_n, y) - d(x_n, y_n)| && \text{by triangle inequality of } (\mathbb{R}, |x - y|) \\
&= |d(x, y) - d(y, x_n)| + |d(y, x_n) - d(x_n, y_n)| && \text{by commutative property of d (Definition 3.1 page 6)} \\
&\le d(x, x_n) + d(y, y_n) && \text{by } 2\sigma = 2^{\frac{1}{p}} \text{ and Lemma 4.9 (page 18)} \\
&= 0 && \text{as } n \to \infty
\end{align*}
$$

<figure><img src="image_76.png" /></figure>

In *distance spaces* and *topological spaces*, limits of convergent sequences are in general *not unique* (Example 3.21 page 12, Example A.17 page 29). However Theorem 4.19 (next) demonstrates that, in a *power distance space*, limits *are* unique.

**Theorem 4.19** (Uniqueness of limit) <sup>47</sup> Let $(X, d, p, \sigma)$ be a POWER DISTANCE SPACE. Let $x, y \in X$ and let $(x_n \in X)$ be an $X$-valued sequence.

$$ \left\{ \begin{array}{l} 1. \quad \{((x_n), (y_n)) \to (x, y)\} \quad \text{and} \\ 2. \quad (p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+ \end{array} \right\} \implies \{x = y\} $$

<figure><img src="image_77.png" /></figure>

PROOF:

(1) lemma: Proof that for all $(p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+$ and for any $\varepsilon \in \mathbb{R}^+$, there exists $N$ such that $d(x, y) < 2\sigma\varepsilon$:

$$
\begin{align*}
d(x, y) &\le \tau(p, \sigma; x, y, x_n) && \text{by definition of power triangle inequality (Definition 4.3 page 16)} \\
&\triangleq 2\sigma \left[ \frac{1}{2} d^p(x, x_n) + \frac{1}{2} d^p(x_n, y) \right]^{\frac{1}{p}} && \text{by definition of power triangle function (Definition 4.1 page 15)} \\
&< 2\sigma \left[ \frac{1}{2} \varepsilon^p + \frac{1}{2} \varepsilon^p \right]^{\frac{1}{p}} && \text{by left hypothesis and for } p \in \mathbb{R}^* \setminus \{-\infty, 0, \infty\} \\
&= 2\sigma\varepsilon \\
d(x, y) &\le \tau(\infty, \sigma; x, y, x_n) && \text{by definition of power triangle inequality at } p = \infty \\
&= 2\sigma \max \{d(x, x_n), d(x_n, y)\} && \text{by Corollary 4.7 (page 17)} \\
&< 2\sigma\varepsilon && \text{by left hypothesis} \\
d(x, y) &\le \tau(-\infty, \sigma; x, y, x_n) && \text{by definition of power triangle inequality at } p = -\infty \\
&= 2\sigma \min \{d(x, x_n), d(x_n, y)\} && \text{by Corollary 4.7 (page 17)} \\
&< 2\sigma\varepsilon && \text{by left hypothesis} \\
d(x, y) &\le \tau(0, \sigma; x, y, x_n) && \text{by definition of power triangle inequality at } p = 0 \\
&= 2\sigma \sqrt{d(x, x_n)} \sqrt{d(x_n, y)} && \text{by Corollary 4.7 (page 17)} \\
&= 2\sigma \sqrt{\varepsilon} \sqrt{\varepsilon} && \text{by left hypothesis} \\
&< 2\sigma\varepsilon && \text{by property of real numbers}
\end{align*}
$$

<figure><img src="image_78.png" /></figure>

<figure><img src="image_79.png" /></figure>

<footnote n="46"><sup>46</sup> in *metric space* ((p, σ) = (1, 1) case): ⇄ [Berberian(1961)] page 37 (Theorem II.4.1)</footnote>

<footnote n="47"><sup>47</sup> in *metric space*: ⇄ [Rosenlicht(1968)] page 46, ⇄ [Thomson et al. (2008) Thomson, Bruckner, and Bruckner] page 32 (Theorem 2.8)</footnote>
(2) Proof that $x = y$ (proof by contradiction):

$$x \neq y \implies d(x, y) \neq 0 \qquad \text{by the nondegenerate property of d (Definition 3.1 page 6)}$$

$$\implies d(x, y) > 0 \qquad \text{by non-negative property of d (Definition 3.1 page 6)}$$

$$\implies \exists \varepsilon \text{ such that } d(x, y) > 2\sigma\varepsilon$$

$$\implies \text{contradiction to (1) lemma page 23}$$

$$\implies d(x, y) = 0$$

$$\implies x = y$$

<figure><img src="image_80.png" /></figure>

## 4.3 Examples

It is not always possible to find a *triangle relation* (Definition 4.3 page 16) $\mathcal{A}(p, \sigma; d)$ that holds in every *distance space* (Definition 3.1 page 6), as demonstrated by Example 4.20 and Example 4.21 (next two examples).

**Example 4.20** Let $d(x, y) \in \mathbb{R}^{\mathbb{R} \times \mathbb{R}}$ be defined such that

$$d(x, y) \triangleq \left\{ \begin{array}{ll} y & \forall (x, y) \in \{4\} \times (0:2] \text{ (vertical half-open interval)} \\ x & \forall (x, y) \in (0:2] \times \{4\} \text{ (horizontal half-open interval)} \\ |x-y| & \text{otherwise} \end{array} \right\}. \quad (\text{Euclidean})$$

Note the following about the pair $(\mathbb{R}, d)$:

(1) By Example 3.21 (page 12), $(\mathbb{R}, d)$ is a *distance space*, but not a *metric space*—that is, the *triangle relation* $\mathcal{A}(1, 1; d)$ does not hold in $(\mathbb{R}, d)$.

(2) Observe further that $(\mathbb{R}, d)$ is *not a power distance space*. In particular, the *triangle relation* $\mathcal{A}(p, \sigma; d)$ does not hold in $(\mathbb{R}, d)$ for any finite value of $\sigma$ (does not hold for any $\sigma \in \mathbb{R}^+$):

$$\begin{align*} d(0, 4) &= 4 \nleq 0 = \lim_{\epsilon \to 0} 2\sigma\epsilon = \lim_{\epsilon \to 0} 2\sigma[\frac{1}{2}|0 - \epsilon|^p + \frac{1}{2}\epsilon^p]^{\frac{1}{p}} \\ &\triangleq \lim_{\epsilon \to 0} 2\sigma[\frac{1}{2}d^p(0, \epsilon) + \frac{1}{2}d^p(\epsilon, 4)]^{\frac{1}{p}} \triangleq \lim_{\epsilon \to 0} \mathcal{A}(p, \sigma; 0, 4, \epsilon; d) \end{align*}$$

**Example 4.21** Let $d(x, y) \in \mathbb{R}^{\mathbb{R} \times \mathbb{R}}$ be defined such that

$$d(x, y) \triangleq \left\{ \begin{array}{ll} |x - y| & \text{for } x = 0 \text{ or } y = 0 \text{ or } x = y \text{ (Euclidean)} \\ 1 & \text{otherwise} \end{array} \right\}. \quad (\text{discrete})$$

Note the following about the pair $(\mathbb{R}, d)$:

(1) By Example 3.22 (page 13), $(\mathbb{R}, d)$ is a *distance space*, but not a *metric space*—that is, the *triangle relation* $\mathcal{A}(1, 1; d)$ does not hold in $(\mathbb{R}, d)$.

(2) Observe further that $(\mathbb{R}, d)$ is *not a power distance space*—that is, the *triangle relation* $\mathcal{A}(p, \sigma; d)$ does not hold in $(\mathbb{R}, d)$ for any value of $(p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+$.

(a) Proof that $\mathcal{A}(p, \sigma; d)$ does not hold for any $(p, \sigma) \in \{\infty\} \times \mathbb{R}^+$:

$$\begin{align*} \lim_{n,m \to \infty} d(1/n, 1/m) &\triangleq 1 \nleq 0 = 2\sigma \max\{0, 0\} && \text{by definition of d} \\ &= 2\sigma \lim_{n,m \to \infty} \max\{d(1/n, 0), d(0, 1/m)\} && \text{by Corollary 4.7 (page 17)} \\ &\geq \lim_{n,m \to \infty} 2\sigma [\frac{1}{2}d^p(1/n, 0) + \frac{1}{2}d^p(0, 1/m)]^{\frac{1}{p}} && \text{by Corollary 4.6 (page 16)} \\ &\triangleq \lim_{n,m \to \infty} \tau(p, \sigma, 1/n, 1/m, 0) && \text{by definition of } \tau \text{ (Definition 4.1 page 15)} \end{align*}$$

<figure><img src="image_81.png" /></figure>

<figure><img src="image_82.png" /></figure>
(b) Proof that $\mathcal{A}(p, \sigma; d)$ does not hold for any $(p, \sigma) \in \mathbb{R}^* \times \mathbb{R}^+$: By Corollary 4.6 (page 16), the triangle function (Definition 4.1 page 15) $\tau(p, \sigma; x, y, z; d)$ is continuous and strictly monotone in $(\mathbb{R}, |\cdot|, \le)$ with respect to the variable $p$. Item 2a demonstrates that $\mathcal{A}(p, \sigma; d)$ fails to hold at the best case of $p = \infty$, and so by Corollary 4.6, it doesn't hold for any other value of $p \in \mathbb{R}^*$ either.

**Example 4.22** Let $d$ be a function in $\mathbb{R}^{\mathbb{R} \times \mathbb{R}}$ such that

$$d(x, y) \triangleq \begin{cases} 2|x - y| & \forall (x, y) \in \{(0, 1), (1, 0)\} & \text{(dilated Euclidean)} \\ |x - y| & \text{otherwise} & \text{(Euclidean)} \end{cases}.$$

Note the following about the pair $(\mathbb{R}, d)$:

(1) By Example 3.23 (page 14), $(\mathbb{R}, d)$ is a distance space, but not a metric space—that is, the triangle relation $\mathcal{A}(1, 1; d)$ does not hold in $(\mathbb{R}, d)$.

(2) But observe further that $(\mathbb{R}, d, 1, 2)$ is a power distance space:

(a) Proof that $\mathcal{A}(1, 2; d)$ (Definition 4.3 page 16) holds for all $(x, y) \in \{(0, 1), (1, 0)\}$:

$$\begin{align*}
d(1, 0) &= d(0, 1) \triangleq 2|0 - 1| = 2 && \text{by definition of d} \\
&\le 2 \le 2(|0 - z| + |z - 1|) && \forall z \in \mathbb{R} \\
&= 2\sigma(\frac{1}{2}|0 - z|^p + \frac{1}{2}|z - 1|^p)^{\frac{1}{p}} && \forall z \in \mathbb{R} \quad \text{for } (p, \sigma) = (1, 2) \\
&\triangleq 2\sigma(\frac{1}{2}d^p(0, z) + d^p(z, 1))^{\frac{1}{p}} && \forall z \in \mathbb{R} \quad \text{for } (p, \sigma) = (1, 2) \text{ and by definition of d} \\
&\triangleq \tau(1, 2; 0, 1, z) && \text{by definition of } \tau \text{ (Definition 4.1 page 15)}
\end{align*}$$

(b) Proof that $\mathcal{A}(1, 2; d)$ holds for all other $(x, y) \in \mathbb{R}^* \times \mathbb{R}^+$:

$$\begin{align*}
d(x, y) \triangleq & 2|x - y| && \text{by definition of d} \\
& \le (|x - z| + |z - y|) && \text{by property of Euclidean metric spaces} \\
& = 2\sigma(\frac{1}{2}|0 - z|^p + \frac{1}{2}|z - 1|^p)^{\frac{1}{p}} && \text{for } (p, \sigma) = (1, 1) \\
& \triangleq \tau(1, 1; x, y, z) && \text{by definition of } \tau \text{ (Definition 4.1 page 15)} \\
& \le \tau(1, 2; x, y, z) && \text{by Corollary 4.6 (page 16)}
\end{align*}$$

(3) In $(X, d)$, the limits of convergent sequences are unique. This follows directly from the fact that $(\mathbb{R}, d, 1, 2)$ is a power distance space (item (2) page 25) and by Theorem 4.19 page 23.

(4) In $(X, d)$, convergent sequences are *Cauchy*. This follows directly from the fact that $(\mathbb{R}, d, 1, 2)$ is a power distance space (item (2) page 25) and by Theorem 4.16 page 21.

**Example 4.23** Let $d$ be a function in $\mathbb{R}^{\mathbb{R} \times \mathbb{R}}$ such that $d(x, y) \triangleq (x - y)^2$. Note the following about the pair $(\mathbb{R}, d)$:

(1) It was demonstrated in Example 3.24 (page 15) that $(\mathbb{R}, d)$ is a distance space, but that it is not a metric space because the triangle inequality does not hold.

(2) However, the tuple $(\mathbb{R}, d, p, \sigma)$ is a power distance space (Definition 4.3 page 16) for any $(p, \sigma) \in \mathbb{R}^* \times [2 : \infty)$: In particular, for all $x, y, z \in \mathbb{R}$, the power triangle inequality (Definition 4.3 page 16) must hold. The “worst case” for this is when a third point $z$ is exactly “halfway between” $x$ and $y$ in $d(x, y)$;

<figure><img src="image_83.png" /></figure>

<figure><img src="image_84.png" /></figure>
that is, when $z = \frac{x+y}{2}$:

$$
\begin{align*}
(x - y)^2 &\triangleq d(x, y) && \text{by definition of } d \\
&\le \tau(p, \sigma; x, y, z; d) && \text{by definition power triangle inequality} \\
&\triangleq 2\sigma \left[ \frac{1}{2}d^p(x, z) + \frac{1}{2}d^p(z, y) \right]^{\frac{1}{p}} && \text{by definition } \tau \text{ (Definition 4.1 page 15)} \\
&\triangleq 2\sigma \left[ \frac{1}{2}(x-z)^{2p} + \frac{1}{2}(z-y)^{2p} \right]^{\frac{1}{p}} && \text{by definition of } d \\
&= 2\sigma \left[ \frac{1}{2}|x-z|^{2p} + \frac{1}{2}|z-y|^{2p} \right]^{\frac{1}{p}} && \text{because } (x)^2 = |x|^2 \text{ for all } x \in \mathbb{R} \\
&= 2\sigma \left[ \left| \frac{1}{2}x - \frac{x+y}{2} \right|^{2p} + \left| \frac{1}{2}y - \frac{x+y}{2} \right|^{2p} \right]^{\frac{1}{p}} && \text{because } z = \frac{x+y}{2} \text{ is the "worst case" scenario} \\
&= 2\sigma \left[ \left| \frac{y-x}{2} \right|^{2p} + \left| \frac{x-y}{2} \right|^{2p} \right]^{\frac{1}{p}} \\
&= 2\sigma \left[ \left| \frac{x-y}{2} \right|^{2p} \right]^{\frac{1}{p}} = \frac{2\sigma}{4} |x-y|^2 \\
&\implies (p, \sigma) \in \mathbb{R}^* \times [2 : \infty)
\end{align*}
$$

(3) The power distance function $d$ is continuous in $(\mathbb{R}, d, p, \sigma)$ for any $(p, \sigma)$ such that $\sigma \ge 2$ and $2\sigma = p^{\frac{1}{p}}$. This follows directly from Theorem 4.18 (page 23).

## Appendix A Topological Spaces

**Definition A.1**<sup>48</sup> Let $\Gamma$ be a set with an arbitrary (possibly uncountable) number of elements. Let $2^X$ be the power set of a set $X$ (Definition 2.2 page 4). A family of sets $T \subseteq 2^X$ is a **topology** on $X$ if

1. $\emptyset \in T$ and

2. $X \in T$ and

3. $U, V \in T \implies U \cap V \in T$ and

4. $\{U_\gamma | \gamma \in \Gamma\} \subseteq T \implies \bigcup_{\gamma \in \Gamma} U_\gamma \in T$.

The ordered pair $(X, T)$ is a topological space if $T$ is a topology on $X$. A set $U$ is **open** in $(X, T)$ if $U$ is any element of $T$. A set $D$ is **closed** in $(X, T)$ if $D^c$ is **open** in $(X, T)$.

Just as the power set $2^X$ and the set $\{\emptyset, X\}$ are algebras of sets on a set $X$, so also are these sets topologies on $X$ (next example):

**Example A.2**<sup>49</sup> Let $\mathcal{T}(X)$ be the set of topologies on a set $X$ and $2^X$ the power set (Definition 2.2 page 4) on $X$.

$\{\emptyset, X\}$ is a topology in $\mathcal{T}(X)$ (indiscrete topology or trivial topology)

$2^X$ is a topology in $\mathcal{T}(X)$ (discrete topology)

<figure><img src="image_85.png" /></figure>

<figure><img src="image_86.png" /></figure>

<footnote n="48"><sup>48</sup> [Munkres(2000)] page 76, [Riesz(1909)], [Hausdorff(1914)], [Tietze(1923)], [Hausdorff(1937)] page 258</footnote>

<footnote n="49"><sup>49</sup> [Munkres(2000)], page 77, [Kubrusly(2011)] page 107 (Example 3.J), [Steen and Seebach(1978)] pages 42–43 (II.4), [DiBenedetto(2002)] page 18</footnote>
**Definition A.3**<sup>50</sup> Let (X, T) be a topological space. A set B ⊆ 2<sup>X</sup> is a **base** for T if

1. $B \subseteq T$ and

2. $\forall U \in T, \exists \{B_\gamma \in B\} \text{ such that } U = \bigcup_\gamma B_\gamma$

**Theorem A.4**<sup>51</sup> Let (X, T) be a TOPOLOGICAL SPACE. Let B be a subset of 2<sup>X</sup> such that B ⊆ 2<sup>X</sup>.

$$ \{ \mathbf{B} \text{ is a BASE for } T \} \iff \left\{ \begin{array}{l} \text{For every } x \in X \text{ and for every OPEN SET } U \text{ containing } x, \\ \text{there exists } B_x \in \mathbf{B} \text{ such that } x \in B_x \subseteq U. \end{array} \right\} $$

**Theorem A.5**<sup>52</sup> Let (X, T) be a TOPOLOGICAL SPACE (Definition A.1 page 26) and B ⊆ 2<sup>X</sup>.

$$ \mathbf{B} \text{ is a base for } (X, T) \iff \left\{ \begin{array}{lcl} 1. & x \in X & \Rightarrow & \exists B_x \in \mathbf{B} \text{ such that } x \in B_x \\ 2. & B_1, B_2 \in \mathbf{B} & \Rightarrow & B_1 \cap B_2 \in \mathbf{B} \end{array} \right. \text{ and} $$

**Example A.6**<sup>53</sup> Let (X, d) be a *metric space*. The set $\mathbf{B} \triangleq \{\mathbf{B}(x, r) | x \in X, r \in \mathbb{N}\}$ (the set of all open balls in (X, d)) is a *base* for a topology on (X, d).

**Example A.7** (the standard topology on the real line)<sup>54</sup> The set $\mathbf{B} \triangleq \{(a : b) | a, b \in \mathbb{R}, a < b\}$ is a *base* for the metric space $(\mathbb{R}, |b-a|)$ (the *usual metric space* on $\mathbb{R}$).

**Definition A.8**<sup>55</sup> Let (X, T) be a topological space (Definition A.1 page 26). Let 2<sup>X</sup> be the power set of X.

The set $A^-$ is the **closure** of $A \in 2^X$ if $A^- \triangleq \bigcap \{D \in 2^X \mid A \subseteq D \text{ and } D \text{ is closed}\}$.

The set $A^\circ$ is the **interior** of $A \in 2^X$ if $A^\circ \triangleq \bigcup \{U \in 2^X \mid U \subseteq A \text{ and } U \text{ is open}\}$.

A point $x$ is a **closure point** of $A$ if $x \in A^-$. A point $x$ is an **interior point** of $A$ if $x \in A^\circ$.

A point $x$ is an **accumulation point** of $A$ if $x \in (A \setminus \{x\})^-$.

A point $x$ in $A^-$ is a **point of adherence** in $A$ or is **adherent** to $A$ if $x \in A^-$. A point $x$ in $A^\circ$ is a **point of adherence** in $A$ or is **adherent** to $A$ if $x \in A^\circ$.

**Proposition A.9**<sup>56</sup> Let (X, T) be a TOPOLOGICAL SPACE (Definition A.1 page 26). Let $A^-$ be the CLOSURE, $A^\circ$ be the INTERIOR, and $\partial A$ be the BOUNDARY of a set A. Let 2<sup>X</sup> be the POWER SET of X.

1. $A^-$ is CLOSED $\forall A \in 2^X$.

2. $A^\circ$ is OPEN $\forall A \in 2^X$.

**Lemma A.10**<sup>57</sup> Let $A^-$ be the CLOSURE, $A^\circ$ the INTERIOR, and $\partial A$ the BOUNDARY of a set A in a topological space (X, T). Let 2<sup>X</sup> be the POWER SET of X.

1. $A^\circ \subseteq A \subseteq A^-$, $\forall A \in 2^X$.

2. $A = A^\circ \iff A \text{ is OPEN}, \forall A \in 2^X$.

3. $A = A^-$, $A \text{ is CLOSED}, \forall A \in 2^X$.

<figure><img src="image_87.png" /></figure>

<figure><img src="image_88.png" /></figure>

<footnote n="50"><sup>50</sup> [Joshi(1983)] page 92 (3.1 Definition), [Davis(2005)] page 46 (Definition 4.15)</footnote>

<footnote n="51"><sup>51</sup> [Joshi(1983)] pages 92–93 (3.2 Proposition), [Davis(2005)] page 46</footnote>

<footnote n="52"><sup>52</sup> [Bollobás(1999)] page 19</footnote>

<footnote n="53"><sup>53</sup> [Davis(2005)] page 46 (Example 4.16)</footnote>

<footnote n="54"><sup>54</sup> [Munkres(2000)] page 81, [Davis(2005)] page 46 (Example 4.16)</footnote>

<footnote n="55"><sup>55</sup> [Gemignani(1972)] pages 55–56 (Definition 3.5.7), [McCarty(1967)] page 90, [Munkres(2000)] page 95 (§Closure and Interior of a Set), [Thron(1966)], pages 21–22 (definition 4.8, definition 4.9), [Kelley(1955)] page 42, [Kubrusly(2001)] pages 115–116</footnote>

<footnote n="56"><sup>56</sup> [McCarty(1967)] page 90 (IV.1 THEOREM)</footnote>

<footnote n="57"><sup>57</sup> [McCarty(1967)] pages 90–91 (IV.1 THEOREM), [ALIPRANTIS AND BURKINSHAW (1998)] page 59</footnote>
**Definition A.11**<sup>58</sup> Let $(X, T_x)$ and $(Y, T_y)$ be topological spaces (Definition A.1 page 26). Let $f$ be a function in $Y^X$. A function $f \in Y^X$ is **continuous** if

$$ \underbrace{U \in T_y}_{\text{open in } (Y, T_y)} \Rightarrow \underbrace{f^{-1}(U) \in T_x}_{\text{open in } (X, T_x)}. $$

A function is **discontinuous** in $(X, T_y)^{(X,T_x)}$ if it is not continuous in $(X, T_y)^{(X,T_x)}$.

<figure><img src="image_89.png" /><figcaption>Figure 3: continuous/discontinuous functions (Example A.12 page 28)</figcaption></figure>

**Example A.12** Some continuous/discontinuous functions are illustrated in Figure 3 (page 28).

Definition A.11 (previous definition) defines continuity using open sets. Continuity can alternatively be defined using closed sets or closure (next theorem).

**Theorem A.13**<sup>59</sup> Let $(X, T)$ and $(Y, S)$ be topological spaces. Let $f$ be a function in $Y^X$.

The following are equivalent:

$$ 1. \text{ f is CONTINUOUS} \iff $$

$$ 2. \text{ B is closed in } (Y, S) \implies f^{-1}(B) \text{ is closed in } (X, T) \quad \forall B \in \mathcal{B}^{(Y)} \iff $$

$$ 3. \text{ } f(A^{-}) \subseteq f(A)^{-} \quad \forall A \in \mathcal{A}^{(X)} \iff $$

$$ 4. \text{ } f^{-1}(B^{-}) \subseteq f^{-1}(B^{-}) \quad \forall B \in \mathcal{B}^{(Y)} $$

**Remark A.14** A word of warning about defining continuity in terms of topological spaces—continuity is defined in terms of a pair of topological spaces, and whether function is continuous or discontinuous in general depends very heavily on the selection of these spaces. This is illustrated in Proposition A.15 (next). The ramification of this is that when declaring a function to be continuous or discontinuous, one must make clear the assumed topological spaces.

**Proposition A.15**<sup>60</sup> Let $(X, T)$ and $(Y, S)$ be TOPOLOGICAL SPACES. Let $f$ be a FUNCTION in $(Y, S)^{(X,T)}$.

$$ 1. \text{ T is the DISCRETE TOPOLOGY} \Rightarrow \text{f is CONTINUOUS} \quad \forall f \in (Y, S)^{(X,T)} $$

$$ 2. \text{ S is the INDISCRETE TOPOLOGY} \Rightarrow \text{f is CONTINUOUS} \quad \forall f \in (Y, S)^{(X,T)} $$

**Definition A.16**<sup>61</sup> Let $(X, T)$ be a topological space (Definition A.1 page 26). A sequence $(x_n)_{n \in \mathbb{Z}}$ converges in $(X, T)$ to a point $x$ if for each open set (Definition A.1 page 26) $U \in T$ that contains $x$ there exists $N \in \mathbb{N}$ such that

<figure><img src="image_90.png" /></figure>

<figure><img src="image_91.png" /></figure>

<footnote n="58"><sup>58</sup> [Davis(2005)] page 34</footnote>

<footnote n="59"><sup>59</sup> [McCarty(1967)] pages 91–92 (IV.2 THEOREM), [SEARCÓID(2006)] PAGE 130 ("THEOREM 8.3.1 (CRITERIA FOR CONTINUITY)", SET IN *metric spaces*)</footnote>

<footnote n="60"><sup>60</sup> [Crossley(2006)] page 18 (Proposition 3.9), [Ponnusamy(2002)] page 98 (2.64. Theorem.)</footnote>

<footnote n="61"><sup>61</sup> [Joshi(1983)] page 83 (3.1 Definition), [Leathem(1905)], page 13 ("→" symbol, section III.11)</footnote>
$x_n \in U$ for all $n > N$.

This condition can be expressed in any of the following forms:

1. The **limit** of the sequence $(x_n)$ is $x$.

2. The sequence $(x_n)$ is **convergent** with limit $x$.

3. $\lim_{n \to \infty} (x_n) = x$.

4. $(x_n) \to x$.

A sequence that converges is **convergent**. A sequence that does not converge is said to **diverge**, or is **divergent**. An element $x \in A$ is a **limit point** of A if it is the limit of some A-valued sequence $(x_n \in A)$.

**Example A.17**$^{62}$ Let $(X, T_{31})$ be a topological space where $X \triangleq \{x, y, z\}$ and
$$T_{31} \triangleq \{\emptyset, \{x\}, \{x, y\}, \{x, z\}, \{x, y, z\}\}.$$

In this space, the sequence $(x, x, x, ...)$ converges to $x$. But this sequence also converges to both $y$ and $z$ because $x$ is in every *open set* (Definition A.1 page 26) that contains $y$ and $x$ is in every *open set* that contains $z$. So, the *limit* (Definition A.16 page 28) of the sequence is *not unique*.

**Example A.18** In contrast to the low resolution topological space of Example A.17, the limit of the sequence $(x, x, x, ...)$ is unique in a topological space with sufficiently high resolution with respect to $y$ and $z$ such as the following: Define a topological space $(X, T_{56})$ where $X \triangleq \{x, y, z\}$ and
$$T_{56} \triangleq \{\emptyset, \{y\}, \{z\}, \{x, y\}, \{y, z\}, \{x, y, z\}\}.$$

In this space, the sequence $(x, x, x, ...)$ converges to $x$ only. The sequence does *not* converge to $y$ or $z$ because there are *open sets* (Definition A.1 page 26) containing $y$ or $z$ that do not contain $x$ (the open sets $\{y\}$, $\{z\}$, and $\{y, z\}$).

**Theorem A.19** (The Closed Set Theorem) $^{63}$ Let $(X, T)$ be a TOPOLOGICAL SPACE. Let $A$ be a subset of $X$ ($A \subseteq X$). Let $A^-$ be the CLOSURE (Definition A.8 page 27) of $A$ in $(X, T)$.

$$ \underbrace{A \text{ is CLOSED in } (X, T)}_{(A = A^-)} \iff \left\{ \begin{array}{l} \text{Every A-valued sequence } (x_n \in A)_{n \in \mathbb{Z}} \\ \text{that CONVERGES in } (X, T) \text{ has its LIMIT in } A \end{array} \right\} $$

**Theorem A.20**$^{64}$ Let $(X, T)$ and $(Y, S)$ be a TOPOLOGICAL SPACES. Let $f$ be a function in $(Y, S)^{(X,T)}$.

$$ \underbrace{\left\{ \begin{array}{l} f \text{ is CONTINUOUS in } (Y, S)^{(X,T)} \\ \text{(Definition A.11 page 28)} \end{array} \right\}}_{\text{INVERSE IMAGE CHARACTERIZATION OF CONTINUITY}} \iff \underbrace{\left\{ \begin{array}{l} ((x_n) \to x \implies f((x_n)) \to f(x) \\ \text{(Definition A.16 page 28)} \end{array} \right\}}_{\text{SEQUENTIAL CHARACTERIZATION OF CONTINUITY}} $$

<figure><img src="image_92.png" /></figure>

PROOF:

(1) Proof for the $\implies$ case (proof by contradiction):

(a) Let $U$ be an *open set* in $(Y, T)$ that contains $f(x)$ but for which there exists no $N$ such that $f(x_n) \in U$ for all $n > N$.

(b) Note that the set $f^{-1}(U)$ is also *open* by the *continuity* hypothesis.

<figure><img src="image_93.png" /></figure>

<figure><img src="image_94.png" /></figure>

<footnote>$^{62}$ [Munkres(2000)] page 98 (Hausdorff Spaces)</footnote>

<footnote>$^{63}$ [Kubrusly(2001)] page 118 (Theorem 3.30), [Haaser and Sullivan(1991)] page 75 (6.9 Proposition),</footnote>

<footnote>[Rosenlicht(1968)] pages 47-48</footnote>

<footnote>$^{64}$ [Ponnusamy(2002)] pages 94-96 ("2.59. Proposition."); in the context of *metric spaces*; includes the "inverse image characterization of continuity" and "sequential characterization of continuity" terminology; this terminology does not seem to be widely used in the literature in general, but has been adopted for use in this text)</footnote>
(c) If $(x_n) \to x$, then

$$f((x_n)) \to f(x) \implies \text{ there exists no } N \text{ such that } f(x_n) \in U \text{ for all } n > N \quad \text{by Definition A.16 (page 28)}$$

$$\implies \text{ there exists no } M \text{ such that } x_n \in f^{-1}(U) \text{ for all } n > M \text{ by definition of } f^{-1}$$

$$\implies (x_n) \to x \quad \text{by continuity hypothesis and def. of convergence (Definition A.16 page 28)}$$

$$\implies \text{ contradiction of } (x_n) \to x \text{ hypothesis}$$

$$\implies f((x_n)) \to f(x)$$

(2) Proof for the $\iff$ case (proof by contradiction):

(a) Let $D$ be a closed set in $(Y, \mathcal{S})$.

(b) Suppose $f^{-1}(D)$ is not closed...

(c) then by the closed set theorem (Theorem A.19 page 29), there must exist a convergent sequence $(x_n)$ in $(X, T)$, but with limit $x$ not in $f^{-1}(D)$.

(d) Note that $f(x)$ must be in $D$. Proof:

(i) by definition of $D$ and $f$, $f((x_n))$ is in $D$

(ii) by left hypothesis, the sequence $f((x_n))$ is convergent with limit $f(x)$

(iii) by closed set theorem (Theorem A.19 page 29), $f(x)$ must be in $D$.

(e) Because $f(x) \in D$, it must be true that $x \in f^{-1}(D)$.

(f) But this is a contradiction to item (2c) (page 30), and so item (2b) (page 30) must be wrong, and $f^{-1}(D)$ must be closed.

(g) And so by Theorem A.13 (page 28), $f$ is continuous.

<figure><img src="image_95.png" /></figure>

# Appendix B Finite sums

## B.1 Convexity

**Definition B.1**$^{65}$

A function $f \in \mathbb{R}^\mathbb{R}$ is **convex** if

$$f(\lambda x + [1 - \lambda]y) \leq \lambda f(x) + (1 - \lambda) f(y) \quad \forall x, y \in \mathbb{R} \text{ and } \forall \lambda \in (0 : 1)$$

A function $g \in \mathbb{R}^\mathbb{R}$ is **strictly convex** if

$$g(\lambda x + [1 - \lambda]y) = \lambda g(x) + (1 - \lambda) g(y) \quad \forall x, y \in D, x \neq y, \text{ and } \forall \lambda \in (0 : 1)$$

A function $f \in \mathbb{R}^\mathbb{R}$ is **concave** if $-f$ is **convex**.

A function $f \in \mathbb{R}^\mathbb{R}$ is **affine** if $f$ is convex and concave.

**Theorem B.2** (Jensen's Inequality)**$^{66}$ Let $f \in \mathbb{R}^\mathbb{R}$ be a function.**

$$\left\{ \begin{array}{ll} \text{1. } f \text{ is } \underset{(\text{Definition B.1 page 30})}{\text{CONVEX}} & \text{and} \\ \text{2. } \sum_{n=1}^{N} \lambda_n = 1 & \text{(WEIGHTS)} \end{array} \right. \implies \left\{ f\left(\sum_{n=1}^{N} \lambda_n x_n\right) \le \sum_{n=1}^{N} \lambda_n f(x_n) \quad \forall x_n \in D, N \in \mathbb{N} \right\}$$

<figure><img src="image_96.png" /></figure>

<figure><img src="image_97.png" /></figure>

<footnote>$^{65}$ [Simon(2011)] page 2, [Barvinok(2002)] page 2, [Bollobás(1999)], page 3, [Jensen(1906)], page 176</footnote>

<footnote>$^{66}$ [Mitrinović et al. (2010) Mitrinović, Pečarić, and Fink] page 6, [Bollobás (1999)] page 3, [Jensen (1906)] pages 179–180</footnote>
## B.2 Power means

**Definition B.3**<sup>67</sup> The $⟨λ_n⟩_1^N$ weighted $\phi$-mean of a tuple $⟨x_n⟩_1^N$ is defined as

$$M_\phi(⟨x_n⟩) \triangleq \phi^{-1}\left(\sum_{n=1}^{N} \lambda_n \phi(x_n)\right)$$

where $\phi$ is a continuous and strictly monotonic function in $\mathbb{R}^{\mathbb{R}^+}$

and $⟨λ_n⟩_1^N$ is a sequence of weights for which $\sum_{n=1}^N λ_n = 1$.

**Lemma B.4**<sup>68</sup> Let $M_\phi(⟨x_n⟩)$ be the $⟨λ_n⟩_1^N$ weighted $\phi$-mean and $M_\psi(⟨x_n⟩)$ the $⟨λ_n⟩_1^N$ weighted $\psi$-mean of a tuple $⟨x_n⟩_1^N$.

$$\phi\psi^{-1} \text{ is CONVEX and } \phi \text{ is INCREASING} \Rightarrow M_\phi(⟨x_n⟩) \ge M_\psi(⟨x_n⟩)$$

$$\phi\psi^{-1} \text{ is CONVEX and } \phi \text{ is DECREASING} \Rightarrow M_\phi(⟨x_n⟩) \le M_\psi(⟨x_n⟩)$$

$$\phi\psi^{-1} \text{ is CONCAVE and } \phi \text{ is INCREASING} \Rightarrow M_\phi(⟨x_n⟩) \le M_\psi(⟨x_n⟩)$$

$$\phi\psi^{-1} \text{ is CONCAVE and } \phi \text{ is DECREASING} \Rightarrow M_\phi(⟨x_n⟩) \ge M_\psi(⟨x_n⟩)$$

One of the most well known inequalities in mathematics is *Minkowski's Inequality*. In 1946, H.P. Mulholland submitted a result that generalizes Minkowski's Inequality to an equal weighted $\phi$-mean.<sup>69</sup> And Milovanović and Milovanovć (1979) generalized this even further to a weighted $\phi$-mean (next).

**Theorem B.5**<sup>70</sup> Let $\phi$ be a function in $\mathbb{R}^R$.

$$\left\{ \begin{array}{ll} 1. & \phi \text{ is CONVEX and } \phi \text{ is STRICTLY MONOTONE and} \\ 2. & \phi \text{ is DECREASING} \\ 3. & \phi(0) = 0 \text{ and } \log \phi \text{ is CONVEX} \end{array} \right. $$

$$\Rightarrow \left\{ \phi^{-1}\left(\sum_{n=1}^{N} \lambda_n \phi(x_n + y_n)\right) \le \phi^{-1}\left(\sum_{n=1}^{N} \lambda_n \phi(x_n)\right) + \phi^{-1}\left(\sum_{n=1}^{N} \lambda_n \phi(y_n)\right) \right\}$$

**Definition B.6**<sup>71</sup> Let $M_{\phi(x;p)}(⟨x_n⟩)$ be the $⟨λ_n⟩_1^N$ weighted $\phi$-mean of a non-negative tuple $⟨x_n⟩_1^N$. A mean $M_{\phi(x;p)}(⟨x_n⟩)$ is a power mean with parameter $p$ if $\phi(x) \triangleq x^p$. That is,

$$M_{\phi(x;p)}(⟨x_n⟩) = \left( \sum_{n=1}^{N} \lambda_n (x_n)^p \right)^{\frac{1}{p}}$$

**Theorem B.7**<sup>72</sup> Let $M_{\phi(x;p)}(⟨x_n⟩)$ be the POWER MEAN with parameter $p$ of an $N$-tuple $⟨x_n⟩_1^N$ in which the elements are NOT all equal.

<figure><img src="image_98.png" /></figure>

<figure><img src="image_99.png" /></figure>

<footnote n="67"><sup>67</sup> [Bollobás(1999)] page 5</footnote>

<footnote n="68"><sup>68</sup> [Pečarić et al.(1992)Pečarić, Proschan, and Tong] page 107, [Bollobás(1999)] page 5, [Hardy et al.(1952)Hardy, Littlewood, and Pólya] page 75</footnote>

<footnote n="69"><sup>69</sup> [Minkowski(1910)] page 115, [Mulholland(1950)], [Hardy et al.(1952)Hardy, Littlewood, and Pólya] (Theorem 24), [Tolsted(1964)] page 7, [Maligranda(1995)] page 258, [Carothers(2000)], page 44, [Bullen(2003)] page 179</footnote>

<footnote n="70"><sup>70</sup> [Milovanović and Milovanović(1979)], [Bullen(2003)] page 306 (Theorem 9)</footnote>

<footnote n="71"><sup>71</sup> [Bullen(2003)] page 175, [Bollobás(1999)] page 6</footnote>

<footnote n="72"><sup>72</sup> [Bullen(2003)] pages 175–177 (see also page 203), [Bollobás(1999)] pages 6–8, [Bullen(1990)] page 250, [Besso(1879)], [Bienaymé(1840)] page 68, [Brenner(1985)] page 160</footnote>
$$ M_{\phi(x;p)} (\langle |x_n| \rangle) \triangleq \left( \sum_{n=1}^{N} \lambda_n (x_n)^p \right)^{\frac{1}{p}} \quad \text{is CONTINUOUS and STRICTLY MONOTONE in } \mathbb{R}^* $$

$$ M_{\phi(x;p)} (\langle |x_n| \rangle) = \begin{cases} \max_{n=1,2,...,N} \langle |x_n| \rangle & \text{for } p = +\infty \\ \prod_{n=1}^{N} x_n^{\lambda_n} & \text{for } p = 0 \\ \min_{n=1,2,...,N} \langle |x_n| \rangle & \text{for } p = -\infty \end{cases} $$

<figure><img src="image_100.png" /></figure>

PROOF:

(1) Proof that $M_{\phi(x;p)}$ is strictly monotone in $p$:

(a) Let $p$ and $s$ be such that $-\infty < p < s < \infty$.

(b) Let $\phi_p \triangleq x^p$ and $\phi_s \triangleq x^s$. Then $\phi_p \phi_s^{-1} = x^{\frac{p}{s}}$.

(c) The composite function $\phi_p \phi_s^{-1}$ is convex or concave depending on the values of $p$ and $s$:

<table><thead><tr><th></th><th>p &lt; 0 (φ<sub>p</sub> decreasing)</th><th>p &gt; 0 (φ<sub>p</sub> increasing)</th></tr></thead><tbody><tr><td>s &lt; 0</td><td>convex</td><td>(not possible)</td></tr><tr><td>s &gt; 0</td><td>convex</td><td>concave</td></tr></tbody></table>

(d) Therefore by Lemma B.4 (page 31),

$$ -\infty < p < s < \infty \quad \Rightarrow \quad M_{\phi(x;p)} (\langle |x_n| \rangle) < M_{\phi(x;s)} (\langle |x_n| \rangle). $$

(2) Proof that $M_{\phi(x;p)}$ is continuous in $p$ for $p \in \mathbb{R} \setminus \{0\}$: The sum of continuous functions is continuous. For the cases of $p \in \{-\infty, 0, \infty\}$, see the items that follow.

(3) Lemma: $M_{\phi(x;-p)} (\langle |x_n| \rangle) = \{M_{\phi(x;p)} (\langle |x_n^{-1}| \rangle)\}^{-1}$. Proof:

$$ \begin{align*} \{M_{\phi(x;p)} (\langle |x_n^{-1}| \rangle)\}^{-1} &= \left\{ \left( \sum_{n=1}^{N} \lambda_n (x_n^{-1})^p \right)^{\frac{1}{p}} \right\}^{-1} && \text{by definition of } M_{\phi} \\ &= \left( \sum_{n=1}^{N} \lambda_n (x_n)^{-p} \right)^{\frac{1}{-p}} && \text{by definition of } M_{\phi} \\ &= M_{\phi(x;-p)} (\langle |x_n| \rangle) \end{align*} $$

(4) Proof that $\lim_{p \to \infty} M_{\phi}(\langle |x_n| \rangle) = \max_{n \in \mathbb{Z}} \langle |x_n| \rangle$:

(a) Let $x_m \triangleq \max_{n \in \mathbb{Z}} \langle |x_n| \rangle$

(b) Note that $\lim_{p \to \infty} M_{\phi} \le \max_{n \in \mathbb{Z}} \langle |x_n| \rangle$ because

$$ \begin{align*} \lim_{p \to \infty} M_{\phi}(\langle |x_n| \rangle) &= \lim_{p \to \infty} \left( \sum_{n=1}^{N} \lambda_n x_n^p \right)^{\frac{1}{p}} && \text{by definition of } M_{\phi} \\ &\le \lim_{p \to \infty} \left( \sum_{n=1}^{N} \lambda_n x_n^p \right)^{\frac{1}{p}} && \text{by definition of } x_m \text{ in item (4a) and because } \phi(x) \triangleq x^p \text{ and } \phi^{-1} \text{ are both increasing or both decreasing} \\ &= \lim_{p \to \infty} \left( x_m^p \sum_{n=1}^{N} \lambda_n \right)^{\frac{1}{p}} && \text{because } x_m \text{ is a constant} \end{align*} $$

<figure><img src="image_101.png" /></figure>

<figure><img src="image_102.png" /></figure>
$$= \lim_{p \to \infty} (x_m^p \cdot 1)^{\frac{1}{p}}$$

$$= x_m$$

$$= \max_{n \in \mathbb{Z}} (|x_n|) \qquad \text{by definition of } x_m \text{ in item (4a)}$$

(c) But also note that $\lim_{p \to \infty} M_{\phi} \ge \max_{n \in \mathbb{Z}} (|x_n|)$ because

$$\lim_{p \to \infty} M_{\phi}(|x_n|) = \lim_{p \to \infty} \left( \sum_{n=1}^{N} \lambda_n x_n^p \right)^{\frac{1}{p}} \qquad \text{by definition of } M_{\phi}$$

$$\ge \lim_{p \to \infty} (w_m x_m^p)^{\frac{1}{p}} \qquad \text{by definition of } x_m \text{ in item (4a) and because } \phi(x) \triangleq x^p \text{ and } \phi^{-1} \text{ are both increasing or both decreasing}$$

$$= \lim_{p \to \infty} w_m^{\frac{1}{p}} x_m^{\frac{p}{p}}$$

$$= x_m$$

$$= \max_{n \in \mathbb{Z}} (|x_n|) \qquad \text{by definition of } x_m \text{ in item (4a)}$$

(d) Combining items (b) and (c) we have $\lim_{p \to \infty} M_{\phi} = \max_{n \in \mathbb{Z}} (|x_n|)$.

$$\text{(5) Proof that } \lim_{p \to -\infty} M_{\phi}(|x_n|) = \min_{n \in \mathbb{Z}} (|x_n|):$$

$$\lim_{p \to -\infty} M_{\phi(x;p)}(|x_n|) = \lim_{p \to \infty} M_{\phi(x:-p)}(|x_n|) \qquad \text{by change of variable } p$$

$$= \lim_{p \to \infty} \{M_{\phi(x;p)}(|x_n^{-1}|)\}^{-1} \qquad \text{by Lemma in item (3) page 32}$$

$$= \lim_{p \to \infty} \frac{1}{M_{\phi(x;p)}(|x_n^{-1}|)}$$

$$= \lim_{p \to \infty} \frac{1}{\lim_{p \to \infty} M_{\phi(x;p)}(|x_n^{-1}|)} \qquad \text{by property of } \lim^{73}$$

$$= \frac{1}{\max_{n \in \mathbb{Z}} (|x_n^{-1}|)} \qquad \text{by item (4)}$$

$$= \frac{1}{\left( \min_{n \in \mathbb{Z}} (|x_n|) \right)^{-1}}$$

$$= \min_{n \in \mathbb{Z}} (|x_n|)$$

$$\text{(6) Proof that } \lim_{p \to 0} M_{\phi}(|x_n|) = \prod_{n=1}^{N} x_n^{\lambda_n} :$$

$$\lim_{p \to 0} M_{\phi}(|x_n|) = \lim_{p \to 0} \exp \left\{ \ln \left\{ M_{\phi}(|x_n|) \right\} \right\}$$

$$= \lim_{p \to 0} \exp \left\{ \ln \left\{ \left( \sum_{n=1}^{N} \lambda_n (x_n^p) \right)^{\frac{1}{p}} \right\} \right\} \qquad \text{by definition of } M_{\phi}$$

<figure><img src="image_103.png" /></figure>

<figure><img src="image_104.png" /></figure>

<footnote n="73">73 [Rudin(1976)] page 85 (4.4 Theorem)</footnote>
$$
\begin{align*}
&= \exp\left\{\frac{\displaystyle\frac{\partial}{\partial p}\ln\left(\sum_{n=1}^{N}\lambda_n(x_n^p)\right)}{\displaystyle\frac{\partial}{\partial p}p}\right\}_{p=0} && \text{by l'Hôpital's rule}^{74} \\
&= \exp\left\{\frac{\displaystyle\sum_{n=1}^{N}\lambda_n\frac{\partial}{\partial p}(x_n^p)}{\displaystyle\sum_{n=1}^{N}\lambda_n(x_n^p)}\right\}_{p=0} &&= \exp\left\{\frac{\displaystyle\sum_{n=1}^{N}\lambda_n\frac{\partial}{\partial p}\exp(\ln(x_n^p))}{\displaystyle\sum_{n=1}^{N}\lambda_n}\right\}_{p=0} \\
&= \exp\left\{\frac{\displaystyle\sum_{n=1}^{N}\lambda_n\frac{\partial}{\partial p}\exp(r\ln(x_n))}{1}\right\}_{p=0} &&= \exp\left\{\displaystyle\sum_{n=1}^{N}\lambda_n\frac{\partial}{\partial p}\exp(p\ln(x_n))\right\}_{p=0} \\
&= \exp\left\{\displaystyle\sum_{n=1}^{N}\lambda_n\exp\{p\ln x_n\}\ln(x_n)\right\}_{p=0} &&= \exp\left\{\displaystyle\sum_{n=1}^{N}\lambda_n\ln(x_n)\right\}_{p=0} \\
&= \exp\left\{\displaystyle\sum_{n=1}^{N}\ln(x_n^{\lambda_n})\right\} &&= \exp\left\{\ln\prod_{n=1}^{N}x_n^{\lambda_n}\right\}_{p=0} = \prod_{n=1}^{N}x_n^{\lambda_n}
\end{align*}
$$

<figure><img src="image_105.png" /></figure>

**Corollary B.8**<sup>75</sup> Let (|x_n|)N be a tuple. Let (|λ_n|)N be a tuple of weighting values such that ΣN_n=1 λ_n = 1.

$$
\min (|x_n|) \le \underbrace{\left( \sum_{n=1}^{N} \lambda_n \frac{1}{x_n} \right)^{-1}}_{\text{harmonic mean}} \le \underbrace{\prod_{n=1}^{N} x_n^{\lambda_n}}_{\text{geometric mean}} \le \underbrace{\sum_{n=1}^{N} \lambda_n x_n}_{\text{arithmetic mean}} \le \max (|x_n|)
$$

<figure><img src="image_106.png" /></figure>

PROOF:

(1) These five means are all special cases of the power mean M<sub>φ(x:p)</sub> (Definition B.6 page 31):

$p = \infty$: $\max(|x_n|)$

$p = 1$: arithmetic mean

$p = 0$: geometric mean

$p = -1$: harmonic mean

$p = -\infty$: $\min (|x_n|)$

(2) The inequalities follow directly from Theorem B.7 (page 31).

<figure><img src="image_107.png" /></figure>

<figure><img src="image_108.png" /></figure>

<footnote n="74">74 ⇔ [Rudin(1976)] page 109 (5.13 Theorem)</footnote>

<footnote n="75">⁷⁵ ⇔ [Bullen(2003)] page 71, ⇔ [Bollobás(1999)] page 5, ⇔ [Cauchy(1821)] pages 457–459 (Note II, theorem 17), ⊠ [Jensen(1906)] page 183, ⊠ [Hoehn and Niven(1985)] page 151</footnote>
(3) Generalized AM-GM inequality: If one is only concerned with the arithmetic mean and geometric mean, their relationship can be established directly using Jensen's Inequality:

$$
\begin{align*}
\sum_{n=1}^{N} \lambda_n x_n &= b^{\log_b (\sum_{n=1}^{N} \lambda_n x_n)} \\
&\ge b^{\left(\sum_{n=1}^{N} \lambda_n \log_b x_n\right)} && \text{by Jensen's Inequality (Theorem B.2 page 30)} \\
&= \prod_{n=1}^{N} b^{(\lambda_n \log_b x_n)} = \prod_{n=1}^{N} b^{(\log_b x_n)\lambda_n} = \prod_{n=1}^{N} x_n^{\lambda_n}
\end{align*}
$$

<figure><img src="image_109.png" /></figure>

B.3 Inequalities

**Lemma B.9** (Young's Inequality) 76

$$
xy < \frac{x^p}{p} + \frac{y^q}{q} \quad \text{with} \quad \frac{1}{p} + \frac{1}{q} = 1 \quad \forall 1 < p < \infty, x, y \ge 0, \text{ but } y \ne x^{p-1}
$$

$$
xy = \frac{x^p}{p} + \frac{y^q}{q} \quad \text{with } \frac{1}{p} + \frac{1}{q} = 1 \quad \forall 1 < p < \infty, x, y \ge 0, \text{ and } y = x^{p-1}
$$

**Theorem B.10** (Minkowski's Inequality for sequences) <fnref n="77" /> Let ($|x_n \in \mathbb{C}|_1$)<fnref n="78" /> and ($|y_n \in \mathbb{C}|_1$)<fnref n="79" /> be complex $N$-tuples.

$$
\left( \sum_{n=1}^{N} |x_n + y_n|^p \right)^{\frac{1}{p}} \le \left( \sum_{n=1}^{N} |x_n|^p \right)^{\frac{1}{p}} + \left( \sum_{n=1}^{N} |y_n|^p \right)^{\frac{1}{p}} \quad \forall 1 < p < \infty
$$

**Appendix C Metric preserving functions**

**Definition C.1** <fnref n="78" /> Let $\mathbb{M}$ be the set of all *metric spaces* (Definition 4.5 page 16) on a set $X$. $\phi \in \mathbb{R}^{\mathbb{R}}$ is a *metric preserving function* if $d(x, y) \triangleq \phi \circ p(x, y)$ is a *metric* on $X$ for all $(X, \mathbb{P}) \in \mathbb{M}$

**Theorem C.2** (necessary conditions) <fnref n="79" /> Let $\mathcal{R}\phi$ be the *RANGE* of a function $\phi$.

$$
\left\{
\begin{array}{l}
\phi \text{ is a} \\
\text{METRIC PRESERVING FUNCTION} \\
(\text{Definition C.1 page 35})
\end{array}
\right\}
\Rightarrow
\left\{
\begin{array}{ll}
1. & \phi^{-1}(0) = \{0\} \qquad \text{and} \\
2. & \mathcal{R}\phi \subseteq \mathbb{R}^{\perp} \qquad \text{and} \\
3. & \phi(x+y) \le \phi(x) + \phi(y) \quad (\phi \text{ is SUBADDITIVE})
\end{array}
\right\}
$$

<figure><img src="image_110.png" /></figure>

<figure><img src="image_111.png" /></figure>

<footnote n="76">⁷⁶ ₩ [Young(1912)] page 226, 🕥 [Hardy et al.(1952)Hardy, Littlewood, and Pólya] <Theorem 24>, ₩ [Tolsted(1964)] page 5, ₩ [Maligranda(1995)] page 257, 🕥 [Carothers(2000)], page 43</footnote>

<footnote n="77">⁷⁷ 🕥 [Minkowski(1910)], page 115, 🕥 [Hardy et al.(1952)Hardy, Littlewood, and Pólya] <Theorem 24>, ₩ [Maligranda(1995)] page 258, 🕥 [Tolsted(1964)], page 7, 🕥 [Carothers(2000)], page 44, 🕥 [Bullen(2003)] page 179</footnote>

<footnote n="78">⁷⁸ 🕥 [Vallin(1999)], page 849 <Definition 1.1>, 🕥 [Corazza(1999)], page 309, 🕥 [Deza and Deza(2009)] page 80</footnote>

<footnote n="79">⁷⁹ 🕥 [Corazza(1999)], page 310 <Proposition 2.1>, 🕥 [Deza and Deza(2009)] page 80</footnote>
**Theorem C.3** (sufficient conditions) <sup>80</sup> Let $\phi$ be a function in $\mathbb{R}^\mathbb{R}$.

$$ \left\{ \begin{array}{ll} 1. & x \ge y \implies \phi(x) \ge \phi(y) \quad \forall x,y \in \mathbb{R}^\mathbb{R} \text{ (ISOTONE)} \\ 2. & \phi(0) = 0 \\ 3. & \phi(x+y) \le \phi(x) + \phi(y) \quad \forall x,y \in \mathbb{R}^\mathbb{R} \text{ (SUBADDITIVE)} \end{array} \right\} \implies \left\{ \begin{array}{l} \phi \text{ is a METRIC} \\ \text{PRESERVING FUNCTION} \\ \text{(Definition C.1 page 35).} \end{array} \right\} $$

<figure><img src="image_112.png" /><figcaption>Figure 4: metric preserving functions</figcaption></figure>

**Example C.4** ($\alpha$-scaled metric/dilated metric) <sup>81</sup> Let $(X, d)$ be a *metric space* (Definition 4.5 page 16).
$\phi(x) \triangleq \alpha x, \alpha \in \mathbb{R}^{+}$ is a *metric preserving function* (Figure 4 page 36 (A)).

PROOF: The proofs for Example C.4-Example C.9 (page 36) follow from Theorem C.3 (page 36).

**Example C.5** (power transform metric/snowflake transform metric) <sup>82</sup> Let $(X, d)$ be a *metric space* (Definition 4.5 page 16).
$\phi(x) \triangleq x^\alpha, \alpha \in (0:1]$, is a *metric preserving function* (see Figure 4 page 36 (B)).

**Example C.6** ($\alpha$-truncated metric/radar screen metric) <sup>83</sup> Let $(X, d)$ be a *metric space* (Definition 4.5 page 16).
$\phi(x) \triangleq \min\{\alpha, x\}, \alpha \in \mathbb{R}^{+}$ is a *metric preserving function* (see Figure 4 page 36 (C)).

**Example C.7** (bounded metric) <sup>84</sup> Let $(X, d)$ be a *metric space* (Definition 4.5 page 16).

$$ \phi(x) \triangleq \frac{x}{1+x} \quad \text{is a metric preserving function (see Figure 4 page 36 (D)).} $$

**Example C.8** (discrete metric preserving function) <sup>85</sup> Let $\phi$ be a function in $\mathbb{R}^\mathbb{R}$.

$$ \phi(x) \triangleq \begin{cases} 0 & \text{for } x \le 0 \\ 1 & \text{otherwise} \end{cases} \quad \text{is a metric preserving function (see Figure 4 page 36 (E)).} $$

**Example C.9** Let $\phi$ be a function in $\mathbb{R}^\mathbb{R}$.

$$ \phi(x) \triangleq \begin{cases} x & \text{for } 0 \le x < 1, \\ x-1 & \text{for } 2 < x < 3, \\ 1 & \text{for } 1 \le x \le 2, \\ 2 & \text{for } x \ge 3 \end{cases} \quad \text{is a metric preserving function (see Figure 4 page 36 (F)).} $$

<figure><img src="image_113.png" /></figure>

<figure><img src="image_114.png" /></figure>

<footnote n="80"><sup>80</sup> [Corazza(1999)] (Proposition 2.3), [Deza and Deza(2009)] page 80, [Kelley(1955)] page 131 (Problem C)</footnote>

<footnote n="81"><sup>81</sup> [Deza and Deza(2006)] page 44</footnote>

<footnote n="82"><sup>82</sup> [Deza and Deza(2009)] page 81, [Deza and Deza(2006)] page 45</footnote>

<footnote n="83"><sup>83</sup> [Giles(1987)], page 33, [Deza and Deza(2006)] pages 242-243</footnote>

<footnote n="84"><sup>84</sup> [Vallin(1999)], page 849, [Aliprantis and Burkinshaw(1998)] page 39</footnote>

<footnote n="85"><sup>85</sup> [Corazza(1999)], page 311</footnote>
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Reference Index

[Aliprantis and Burkinshaw (1998)], 6, 19, 21, 27, 36
[Adams and Franzosa (2008)], 8
[Apostol (1975)], 5, 9, 21
[Barvinok (2002)], 30
[Beran (1985)], 5
[Berberian (1961)], 18, 23
[Bessenyei and Pales (2014)], 6, 16
[Besso (1879)], 31
[Bienaymé (1840)], 31
[Blumenthal (1938)], 6
[Blumenthal (1953)], 6, 11-15
[Bollobás (1999)], 27, 30, 31, 34
[Bottazzini (1986)], 4
[Bourbaki (1939)], 4
[Brenner (1985)], 31
[Brown and Watson (1996)], 8

[Bruckner et al. (1997) Bruckner, Bruckner, and Thomson], 6
[Brunschwig et al. (2003) Brunschwig, Lloyd, and Pellegrin], 15
[Bryant (1985)], 10
[Bullen (1990)], 31
[Bullen (2003)], 31, 34, 35
[Burstein et al. (2005) Burstein, Ulitsky, Tuller, and Chor], 2
[Carothers (2000)], 31, 35
[Cauchy (1821)], 34
[Chatterji (1967)], 8
[Choquet (1954)], 5
[Cohn (2002 December 6)], 5
[Comtet (1966)], 8
[Comtet (1974)], 4, 8
[Copson (1968)], 6, 16
[Corazza (1999)], 35, 36

[Costa et al. (2004) Costa, Castro, Rowstron, and Key], 2
[Crammer et al. (2007) Crammer, Kearns, and Wortman], 2
[Crossley (2006)], 28
[Czerwik (1993)], 16
[Davis (2005)], 11, 27, 28
[Dedekind (1900)], 5
[Deza and Deza (2006)], 36
[Deza and Deza (2009)], 35, 36
[Deza and Deza (2014)], 6, 16
[DiBenedetto (2002)], 26
[Dieudonné (1969)], 7, 16, 18
[Euclid (circa 300BC)], 16
[Evans et al. (1967) Evans, Harary, and Lynn], 8
[Fagin et al. (2003a) Fagin, Kumar, and Sivakumar], 16

<figure><img src="image_129.png" /></figure>

<figure><img src="image_130.png" /></figure>
[Fagin et al.(2003b)Fagin, Kumar, and Sivakumar], 16
[Fréchet(1906)], 16
[Fréchet(1928)], 16
[Galvin and Shore(1984)], 6
[Gemignani(1972)], 27
[Gibbons et al.(1977)Gibbons, Olkin, and Sobel], 16
[Giles(1987)], 9, 21, 36
[Haaser and Sullivan(1991)], 10, 29
[Hahn and Rosenthal(1948)], 4, 5
[Halmos(1950)], 5
[Halmos(1960)], 4
[Hardy et al.(1952)Hardy, Littlewood, and Pólya], 31, 35
[Hausdorff(1914)], 26
[Hausdorff(1937)], 6, 11, 16, 26
[Heath(1961)], 6
[Heinonen(2001)], 16
[Hoehn and Niven(1985)], 16, 34
[Jensen(1906)], 30, 34
[Jiménez and Yukich(2006)], 2
[Joshi(1983)], 8, 27, 28
[Kelley(1955)], 4, 27, 36
[Khamsi and Kirk(2001)], 6, 9
[Kirk and Shahzad(2014)], 16
[Korselt(1894)], 5
[Krishnamurthy(1966)], 8

[Kubrusly(2001)], 10, 27, 29
[Kubrusly(2011)], 26
[Laos(1998)], 6, 15
[Leathem(1905)], 9, 28
[Lin(1998)], 2
[MacLane and Birkhoff(1999)], 5
[Maddux(2006)], 4
[Maligranda and Orlicz(1987)], 2
[Maligranda(1995)], 31, 35
[McCarty(1967)], 27, 28
[Menger(1928)], 6
[Michel and Herget(1993)], 6, 18
[Milovanović and Milovanović(1979)], 31
[Minkowski(1910)], 31, 35
[Mitrinović et al.(2010)Mitrinović, Pečarić, and Fink], 30
[Molchanov(2005)], 4, 6
[Mulholland(1950)], 31
[Munkres(2000)], 26, 27, 29
[Murdeshwar(1990)], 8
[Ore(1935)], 5
[Pap(1995)], 4, 5
[Pečarić et al.(1992)Pečarić, Proschan, and Tong], 31
[Ponnusamy(2002)], 28, 29
[Rana(2002)], 4
[Riesz(1909)], 26

[Rosenlicht(1968)], 7, 9, 10, 21–23, 29
[Rudin(1976)], 33, 34
[Salzmann et al.(2007)Salzmann, Grundhöfer, Hähl, and Löwen], 8
[Schweizer and Sklar(1983)], 16
[Searcóid(2006)], 28
[Sherstnev(1962)], 16
[Simon(2011)], 30
[Steen and Seebach(1978)], 26
[Suppes(1972)], 4
[Sutherland(1975)], 10
[Szirmai(2007)], 2
[Thomson et al.(2008)Thomson, Bruckner, and Bruckner], 23
[Thron(1966)], 6, 27
[Tietze(1923)], 26
[Tolsted(1964)], 31, 35
[Vallin(1999)], 35, 36
[Veltkamp and Hagedoorn(2000)], 2
[Veltkamp(2001)], 2
[Vitányi(2011)], 2
[Walmsley(1920)], 15
[Wilson(1931)], 6
[Xia(2009)], 16
[Young(1912)], 35
[Zorich(2004)], 19

Subject Index

α-scaled, 36

α-scaled metric, 36

α-truncated, 36

α-truncated metric, 36

φ-mean, 31

σ-inframetric inequality, 3, 16, 16

σ-inframetric space, 3, 16

absolute value, 5, 5, 12, 18

accumulation point, 3, 4, 27

adherent, 27

affine, 17, 30

AM-GM inequality, 34

anti-symmetric, 5

arithmetic mean, 1, 3, 17

arithmetic mean geometric mean inequality, 34

b-metric, 16

base, 3, 4, 6, 8, 9, 12–14, 20, 27, 27
boundary, 27
bounded, 2, 6, 9, 10, 21, 22, 36
bounded metric, 36

Cantor Intersection Theorem, 2
Cantor intersection theorem, 11, 11
cardinality, 5
cardinality of A, 5
Cartesian product, 4
Cauchy, 2, 3, 6, 9, 10, 13, 14, 21, 22, 25
Cauchy condition, 9
Cauchy sequence, 9
Cauchy sequences, 9
closed, 7, 7, 10, 11, 26, 27, 29,

30
closed ball, 6
closed interval, 5
closed set, 3, 4, 7
closed set theorem, 30
closure, 3, 4, 10, 27, 27, 29
closure point, 27
commutative, 18, 23
commutative ring, 5
complete, 9, 10, 11
concave, 30, 30, 31, 32
continuity, 3, 4, 12, 15, 17, 28–30
continuous, 1, 3, 6, 11, 11, 12, 15, 17, 23, 25, 26, 28, 28, 28, 29–32
contradiction, 24
converge, 9
convergence, 3, 4, 13, 22, 30
metric space, 21

<figure><img src="image_131.png" /></figure>

<figure><img src="image_132.png" /></figure>
convergent, 3, 6, 9, **9**, 10–14, 21, 22, 25, **29**, 30

convergent sequence, 3

converges, **28**, 29

convex, 30, **30**, 31, 32

functional, 30

strictly, 30

decreasing, 31

definitions

σ-inframetric space, **16**

accumulation point, **27**

base, **27**

Cartesian product, **4**

Cauchy sequence, **9**

closed ball, **6**

closed interval, **5**

closed set, **7**

closure point, **27**

distance space, **6**

empty set, **5**

extended real numbers, **4**

function, **4**

half-open interval, **5**

inframetric space, **16**

interior, **27**

interior point, **27**

metric space, **16**

near metric space, **16**

open ball, **6**

open interval, **5**

open set, **7**

ordered pair, **4**

ordered set, **5**

partially ordered set, **5**

paving, **4**

point of adherence, **27**

poset, **5**

power distance space, **16**

relation, **4**

set function, **5**

set of all functions, **4**

set of all relations, **4**

set of integers, **4**

set of natural numbers, **4**

set of non-negative real

numbers, **4**

set of positive real num-

bers, **4**

set of real numbers, **4**

set structure, **4**

topological space in-
duced by (X, d), **8**

topology, **26**

topology induced by
(X, d) on X, **8**

diameter, **6**

dilated, 36

dilated Euclidean, 14, 25

dilated metric, 36

discontinuous, **11**, 12–15, 28, **28**

discrete, 13, 24, 36

discrete metric preserving
function, **36**

discrete topology, 26, 28

distance, 6, **6**, 12–15

distance function, 11, 12, 14, 15

distance space, 1–6, **6**, 7–17, 21, 23–25

distance spaces, 3

diverge, 14, **29**

divergent, **29**

empty set, **5**, 7

Euclidean, 12–14, 24, 25

Euclidean metric space, 25

examples

α-scaled, 36

α-scaled metric, 36

α-truncated, 36

α-truncated metric, **36**

bounded, 36

bounded metric, 36

dilated, 36

dilated metric, 36

discrete, 36

discrete metric preserving
function, **36**

power transform, 36

power transform metric,

36

radar screen, **36**

radar screen metric, **36**

snowflake, 36

snowflake transform met-
ric, 36

the standard topology on
the real line, **27**

extended real numbers, **4**, 15

extended set of integers, 4

finite, 5

function, **4**, 28

functions

absolute value, 5, **5**, 12, 18

arithmetic mean, 17

cardinality of A, 5

distance, 6, **6**, 12–15

distance function, 11, 14,

15

geometric mean, 17

harmonic mean, 17

maximum, 17

metric, 14

metric function, 3, 12

metric preserving func-
tion, 35, **35**, 36

minimum, 17

modulus, **5**

power distance, **16**

power distance function,
**16**, 17, 26

Power mean, 3, 15

power mean, 1, 3, 15, 31,

**31**

power triangle function,
1, 3, 15, **15**, 16, 17, 22, 23

quadratic mean, 17

sequence, 6, 9–11

set function, 5

triangle function, 16, 25

Generalized AM-GM inequal-
ity, 35

generalized arithmetic mean

geometric mean inequality, 34

geometric inequality, **16**, 18

geometric mean, 1, 3, 17

half-open interval, **5**, 12

harmonic inequality, **16**, 18

harmonic mean, 1, 3, 17

homogeneous, 5

horizontal half-open interval,
12, 24

increasing, 31

indiscrete topological space, 8

indiscrete topology, 26, 28

inequalities

AM-GM, 34

Jensen's, 30

Minkowski (sequences),

35

power triangle inequality,

23

Young, 35

inframetric inequality, 1, 16, **16**, 18

inframetric space, **16**, 17

interior, 3, 4, 27, **27**

interior point, **27**

inverse image characteriza-
tion, 12

inverse image characterization
of continuity, 12, 29

isotone, 36

Jensen's Inequality, **30**, 35

l'Hôpital's rule, 34

<figure><img src="image_133.png" /></figure>

<figure><img src="image_134.png" /></figure>
limit, 9, 10, 29, 29

limit point, 29

linearly ordered, 8

maximum, 1, 3, 17

mean square, 1, 3

metric, 14, 35

metric function, 3, 12

metric preserving function, 35, 35, 36

Metric space, 1, 2

metric space, 2–16, **16**, 18, 21–25, 27–29, 35, 36

metrics

α-scaled metric, 36

α-truncated metric, 36

bounded, 36

dilated metric, 36

power transform metric,

36

radar screen, 36

minimal inequality, **16**, 18

minimum, 1, 3, 17

Minkowski's Inequality, 31

Minkowski's Inequality for sequences, 35

modulus, 5

monontone, 3

monotone, 1, 19

near metric space, 3, 16, **16**, 17

non-negative, 5, 6, 18, 24, 31

nondegenerate, 5, 6, 19, 24

not Cauchy, 14

not closed, 30

not open, 12, 13

not unique, 13, 23, 29

open, 2–4, 6, 7, **7**, 8, 9, 12–14, 20, 21, 26, **26**, 27–29

open ball, 3, 4, 6, **6**, 7–9, 12–14, 19–21

open balls, 12, 20

open interval, 5, 12

open set, 7, **7**, 8, 12, 13, 20, 27–29

operations

φ-mean, 31

arithmetic mean, 1, 3

closure, 27

geometric mean, 1, 3

harmonic mean, 1, 3

limit, 9, 29

maximum, 1, 3

mean square, 1, 3

minimum, 1, 3

open set, 7

set difference, 7

set intersection, 7

SMR, 16

square-mean-root, 16

order relation, 5, 5

ordered metric space, 16

ordered pair, 4, 12

ordered set, 5, 5

ordered set of real numbers, 5

partial order relation, 5

partially ordered set, 5

paving, 4, 5

point of adherence, 27

poset, 5

positive real numbers, 15

power distance, 16

power distance function, 16, 17, 26

power distance space, 3, 4, **16**, 17, 19–25

power distance spaces, 3

Power mean, 3, 15

power mean, 1, 3, 15, 31, **31**, 34

power set, 4, 6, 26, 27

power set of X, 4

power transform, 36

power transform metric, 36

power triangle function, 1, 3, 15, **15**, 16, 17, 22, 23

power triangle inequality, 1, 3, 15, 16, **16**, 17, 18, 21–23, 25, 26

power triangle triangle space, 18

preorder, 5

probabilistic metric spaces, 16

properties

adherent, 27

affine, 17, 30

anti-symmetric, 5

bounded, 2, **6**, 9, 10, 21, 22

Cauchy, 2, 3, 6, 9, 10, 13,

14, 21, 22, 25

Cauchy condition, 9

closed, 7, **7**, 10, 11, **26**, 27,

29, 30

closed set, 4

commutative, 18, 23

complete, 9, 10, 11

concave, 30, **30**, 31, 32

continuity, 3, 4, 12, 15, 17,

28–30

continuous, 1, 3, 6, 11, **11**,

12, 15, 17, 23, 25, 26, 28, **28**, 29–32

converge, 9

convergence, 3, 4, 13, 22,

30

convergent, 3, 6, 9, **9**, 10–14, 21, 22, 25, **29**, 30

converges, 28, 29

convex, 30, **30**, 31, 32

decreasing, 31

dilated Euclidean, 14, 25

discontinuous, **11**, 12–15,

28, **28**

discrete, 13, 24

distance space, 11

diverge, 14, 29

divergent, 29

Euclidean, 12–14, 24, 25

finite, 5

homogeneous, 5

increasing, 31

isotone, 36

limit, 9

linearly ordered, 8

monontone, 3

monotone, 1, 19

non-negative, 5, 6, 18, 24,

31

nondegenerate, 5, 6, 19,

24

not Cauchy, 14

not closed, 30

not open, 12, 13

not unique, 13, 23, 29

open, 2, 3, 6, 7, **7**, 8, 9, 12–14, 20, 21, 26, **26**, 27, 28

open set, 8

power triangle inequality, 17, 18, 26

reflexive, 5

strictly convex, **30**

strictly monontone, 15

strictly monotone, 10, 17,

18, 22, 25, 31, 32

strictly monotonic, 31

subadditive, 5, 35, 36

submultiplicative, 5

symmetric, 6, 18, 19, 22

transitive, 5

triangle inequality, 2, 3, 5,

12–15, 23, 25

unique, 3, 6, 10, 12–14, 25,

29

quadratic inequality, **16**

quadratic mean, 17

quadratic mean inequality, 18

radar screen, 36

radar screen metric, 36

range, 35

<figure><img src="image_135.png" /></figure>

<figure><img src="image_136.png" /></figure>
real line, 8

reflexive, 5

relation, 4, 4, 16

relations

σ-inframetric inequality,
3, 16, **16**

geometric inequality, **16**

harmonic inequality, **16**

inframetric inequality, 1,

16, **16**

minimal inequality, **16**

order relation, 5

partial order relation, 5

power triangle inequality,

1, 3, 15, 16, **16**, 21, 22, 25

preorder, 5

quadratic inequality, **16**

relaxed triangle inequal-

ity, 1, 3, 16, **16**

square mean root in-
equality, **16**

triangle inequality, 1, 2,

15, 16, **16**

triangle relation, 24, 25

relaxed triangle inequality, 1, 3,

16, **16**

sequence, 6, 9–11

sequences

Cauchy, 9

sequential characterization, 12

sequential characterization of
continuity, 12, 29

set, 5, 7

set difference, 7

set function, 5, 5

set intersection, 7

set of all functions, 4

set of all relations, 4

set of extended real numbers, 3

set of integers, 4

set of natural numbers, 4

set of non-negative real num-
bers, 4

set of positive real numbers, 3,
6

set of positive real numbers, 4

set of real numbers, 4

set structure, 4

sets, 4

SMR, 16

snowflake, 36

snowflake transform metric, 36

space

metric, 7

topological, 26

square mean root (SMR) in-

equality, 18

square mean root inequality,
**16**

square-mean-root, 16

strictly convex, 30, **30**

strictly monontone, 15

strictly monotone, 10, 17, 18,
22, 25, 31, 32

strictly monotonic, 31

structures

σ-inframetric space, 3, **16**
accumulation point, 3, 4,
**27**

b-metric, 16

base, 3, 4, 6, 8, 9, 12–14,
20, 27, **27**

boundary, 27

Cartesian product, 4

Cauchy sequence, 9

closed ball, 6

closed interval, 5

closed set, 3, 7

closure, 3, 4, 10, 27, 29

closure point, **27**

commutative ring, 5

convergent sequence, 3

discrete topology, 26, 28

distance function, 12

distance space, 1–6, **6**, 7–

17, 21, 23–25

distance spaces, 3

empty set, 5, 7

Euclidean metric space,

25

extended real numbers, 4,

15

extended set of integers, 4

function, 4, 28

geometric inequality, 18

half-open interval, 5, 12

harmonic inequality, 18

horizontal half-open in-
terval, 12, 24

indiscrete topological
space, 8

indiscrete topology, 26, 28

inframetric inequality, 18

inframetric space, **16**, 17

interior, 3, 4, 27, **27**

interior point, **27**

limit, 29

metric, 35

Metric space, 1, 2

metric space, 2–16, **16**, 18,

21–25, 27–29, 35, 36

minimal inequality, 18

near metric space, 3, 16,
**16**, 17

open, 4, 29

open ball, 3, 4, 6, **6**, 7–9,
12–14, 19–21

open balls, 12, 20

open interval, 5, 12

open set, 7, **7**, 12, 13, 20,
27–29

order relation, 5

ordered metric space, 16

ordered pair, 4, 12

ordered set, 5, 5

ordered set of real num-
bers, 5

partially ordered set, 5

paving, 4, 5

point of adherence, **27**

poset, 5

positive real numbers, 15

power distance space, 3,
4, **16**, 17, 19–25

power distance spaces, 3

power set, 4, 6, 26, 27

power set of X, 4

power triangle inequality,
3

power triangle triangle
space, 18

quadratic mean inequal-
ity, 18

range, 35

real line, 8

relation, 4, 4, 16

set, 5, 7

set function, 5

set of all functions, 4

set of all relations, 4

set of extended real num-
bers, 3

set of integers, 4

set of natural numbers, 4

set of non-negative real
numbers, 4

set of positive real num-
bers, 3, 6

set of positive real num-
bers, 4

set of real numbers, 4

set structure, 4

sets, 4

square mean root (SMR)

inequality, 18

topological space, 3, 4, 8,
12, 23, 26–29

topological space (X, T)

<figure><img src="image_137.png" /></figure>

<figure><img src="image_138.png" /></figure>
induced by (ℝ, d), 13
topological space in-
duced by (ℝ, d), 12
topological space in-
duced by (X, d), 8, **8**, 10, 21
topologogical space, 8
topology, 3, 4, 8, 12–14,
26, **26**
topology induced by
(X, d) on X, **8**
triangle inequality, 2, 18
triangle relation, 16
trivial topology, 26
usual metric space, 27
vertical half-open inter-
val, 12, 24
weights, 30
subadditive, 5, 35, 36
submultiplicative, 5
symmetric, 6, 18, 19, 22

The Archimedean Property, 19
The Closed Set Theorem, **29**
The principle of Archimedes,
19
the standard topology on the
real line, **27**
theorems
Cantor intersection, 11
Cantor Intersection Theo-
rem, 2
Cantor intersection theo-
rem, **11**

closed set theorem, 30
Generalized AM-GM in-
equality, 35
inverse image characteri-
zation, 12
inverse image characteri-
zation of continuity, 12, 29
Jensen's Inequality, **30**, 35
l'Hôpital's rule, 34
Minkowski's Inequality,
31
Minkowski's Inequality
for sequences, **35**
sequential characteriza-
tion, 12
sequential characteriza-
tion of continuity, 12, 29
The Archimedean Prop-
erty, 19
The Closed Set Theorem,
**29**
The principle of
Archimedes, 19
Uniqueness of limit, **23**
Young's Inequality, **35**
topological space, 3, 4, 8, 12,
23, 26–29
topological space (X, T) in-
duced by (ℝ, d), 13
topological space induced by
(ℝ, d), 12
topological space induced by

(X, d), 8, **8**, 10, 21
topologies
discrete, 26
indiscrete, 26
trivial, 26
topologogical space, 8
topology, 3, 4, 8, 12–14, 26, **26**
topology induced by (X, d) on
X, **8**
transitive, 5
triangle function, 16, 25
triangle inequality, 1–3, 5, 12–
16, **16**, 18, 23, 25
triangle relation, 16, 24, 25
trivial topology, 26
unique, 3, 6, 10, 12–14, 25, 29
Uniqueness of limit, **23**
usual metric space, 27
values
cardinality, 5
contradiction, 24
diameter, **6**
limit, 10
limit point, **29**
vertical half-open interval, 12,
24
weighted, 31
weights, 30
Young's Inequality, **35**

Telecommunications Engineering Department, National Chiao-Tung University, Hsinchu, Taiwan; 國立交通大學
(Gáo Lì Jiào Tōng Dà Xúe) 電信工程學系 (Diàn Xìn Gōng Chéng Xúe Xì) 新竹, 台灣 (Xīn Zhú, Tái Wān)

dgreenhoe@gmail.com

Received: aa bb 20YY Revised: cc dd 20ZZ

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