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THE EMPEROR IN NEW CLOTHES:
COMPACTIFICATION OF DESCRIPTIVE
HAUSDORFF TOPOLOGICAL SPACES.
APPLICATION: SATURN RING CHARGED WATER
MOLECULES

James F Peters, Sheela Ramanna, Alkan A.O. Ozkan

► To cite this version:

James F Peters, Sheela Ramanna, Alkan A.O. Ozkan. THE EMPEROR IN NEW CLOTHES: COMPACTIFICATION OF DESCRIPTIVE HAUSDORFF TOPOLOGICAL SPACES. APPLICATION: SATURN RING CHARGED WATER MOLECULES. 2024. hal-04712147

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Preprint submitted on 2 Oct 2024

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THE EMPEROR IN NEW CLOTHES:
COMPACTIFICATION OF DESCRIPTIVE HAUSDORFF
TOPOLOGICAL SPACES. APPLICATION: SATURN RING
CHARGED WATER MOLECULES

ALKAN ÖZKAN, JAMES F. PETERS, METIN DUMAN, AND SHEELA RAMANNA

ABSTRACT. In this paper, we introduce the emperor in new clothes, namely, compactification of a descriptive Hausdorff topological space, which resides in the Euclidean topological space, which is non-compactifiable. A descriptive Hausdorff topological space provide an effective means of measuring the proximities of object descriptions as well as providing a framework for novel compactifiable topological spaces. The main results of this paper are (1) There is a descriptive Hausdorff topology on every descriptive Hausdorff proximity space (Theorem 2.4) and (2) Every descriptive Hausdorff topological space is locally compact (Theorem 3.4).

CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Preliminaries</td><td>2</td></tr><tr><td>3. Compactification of a Descriptive Hausdorff Topological Space</td><td>5</td></tr><tr><td>4. Application: Saturn Ring Charged Water Molecules</td><td>6</td></tr><tr><td>5. Conclusion</td><td>7</td></tr><tr><td>References</td><td>7</td></tr></table>

1. INTRODUCTION

This paper introduces the emperor of topology (compactification) in new clothes. This is accomplished by taking a non-compact Euclidean space and modifying it by embedding in it a locally compact space, namely, a descriptive Hausdorff topological space. This is made possible by the introduction of descriptively proximal sets in a proximity space [11] as well as recent work on the topology of descriptively proximal sets [6], [14] in addition and descriptively near soft sets [8],[12] [1],[7],[9]. The results in this paper to a simpler form of proximal topological spaces that are more amenable in applications (see, e.g., [13]). An important bi-product of this work is the compactification of a descriptive Hausdorff topological space that is locally compact in Euclidean space.

<footnote n="2020">2020 Mathematics Subject Classification. 054E05 (Proximity); 54C50 (Topology).</footnote>

<footnote>Key words and phrases. Compactification, Description, Descriptively near, Hausdorff distance, Proximity, Soft set.</footnote>
Let $A \in 2^X$ for a nonempty set $A$ with $n$ characteristics, $A_i \in 2^A$. A characteristic of a subset $A_i$ is a mapping $\varphi: 2^A \to \mathbb{R}$ defined by $\varphi(A_i) \in \mathbb{R}$. A complete description of $A$ with $n$ characteristics is a mapping $\Phi: 2^A \to 2^{\mathbb{R}^n}, n \ge 1$ defined by $\Phi(A) = \{\varphi(A_1), \varphi(A_2), \dots, \varphi(A_i), \dots, \varphi(A_n)\}, A_i \in 2^A, n \ge 1$.

**Definition 1.** Descriptively Proximal Sets [6]. Let $X$ be a nonempty set, $2^X$ the collection of subsets of $X$, $A, B \in 2^X$. Next, for $\Phi(A) \cap \Phi(B) \neq \emptyset$, consider the descriptive proximity mapping $\delta_{\Phi}: 2^X \times 2^X \to \mathbb{R}$ defined by

$$|\varphi(A_i) - \varphi(B_i)| = r \in \mathbb{R}^{0+}, A_i \in 2^A, B_i \in 2^B, i \ge 1$$

i.e., $A, B$ are descriptively proximal, provided $\delta_{\Phi}(A, B) > 0$ for at least one pair of characteristics $\phi(A_i) \in \Phi(A), \phi(B_i) \in \Phi(B)$. in the description of $A$ and $B$ are identical, i.e., $|\varphi(A) - \varphi(B)| = 0$. ■

**Remark 1.** The total number of characteristics of a physical set is usually not known. As a result, the description of a physical set is indefinite (see Axiom 1) and the descriptions of every pair of physical objects is not fixed (see Axiom 2). ■

**Axiom 1.** Let $\Phi(A)$ be a set of characteristics of a nonempty set of physical objects $A$ and let $|\Phi(A)|$ be the size of $\Phi(A)$. The number of characteristics of $A$ is indefinite, i.e., $|\Phi(A)| = k \in \mathbb{Z}^+$.

**Axiom 2.** For each pair of sets of physical objects $A$ and $B$, the difference between the sizes of the descriptions is not fixed, i.e., $|\Phi(A) - \Phi(B)| = r \in \mathbb{R}^{0+}$.

A list of the principal symbols used in this paper is given in Table 1.

<table><caption>TABLE 1. Principal Symbols Used in this Paper</caption><thead><tr><th>Symbol</th><th>Meaning</th></tr></thead><tbody><tr><td>φ(A)</td><td>Single characteristic of a nonempty set A</td></tr><tr><td>Φ(A)</td><td>Description of a nonempty set A</td></tr><tr><td>δ<sub>Φ</sub>(A, B)</td><td>Descriptively proximal sets A, B</td></tr><tr><td>δ<sub>limΦ</sub>(A, B)</td><td>Descriptively proximal soft sets A, B</td></tr><tr><td>d<sub>H</sub>(A, B)</td><td>Hausdorff Distance between A, B</td></tr><tr><td>d<sub>H</sub>(Φ(A), Φ(B))</td><td>Descriptive Hausdorff Distance between A, B</td></tr><tr><td>τ<sub>Φ</sub></td><td>Descriptive Topology</td></tr><tr><td>τ<sub>limΦ</sub></td><td>Descriptive soft Topology</td></tr><tr><td>A</td><td>Open Set</td></tr><tr><td>F = 2<sup>A</sup></td><td>Open Cover</td></tr></tbody></table>

## 2. PRELIMINARIES

This section introduces **proximal topological spaces** in terms of descriptive proximity.

**Definition 2.** Descriptively Proximal Sets [10, 6]. Let $X$ be a nonempty set, $2^X$ the collection of subsets of $X$, $A, B \in 2^X$. Next, consider the descriptive proximity mapping $\delta_{\Phi}$:
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Torus near sets</figcaption></figure>

$$ \delta_{\Phi}: 2^{\mathbf{X}} \times 2^{\mathbf{X}} \to \mathbb{R} \text{ defined by } \delta_{\Phi}(A, B) = |\Phi(A) - \Phi(B)| = r \in \mathbb{R}^{0,+}. $$

i.e., A, B are descriptively proximal, provided $\delta_{\Phi}(A, B) \ge 0$. The descriptions of A and B are identical, provided $\delta_{\Phi}(A, B) = 0$. ■

**Example 1.** Let $\varphi(A) = k \text{ nm (nanometers)} \in \mathbb{R}$ be the description of a torus panel shown in Fig. 1, limited to a single characteristic, namely, a panel color wavelength in the visible spectrum. Assume each gray panel has a wavelength = 304 nm. Consider the rectangular panels A, B, C, E, H on the surface of the torus in Fig. 1. Notice that

$$ \varphi(A) = \varphi(B) = \varphi(H) = 304 \text{ nm}. $$

In addition, we have following descriptive proximities:

1º $A \delta_{\Phi} E$ (panel A not descriptively near panel E).

2º $A \delta_{\Phi} C$ (panel A not descriptively near panel C).

3º $A \delta_{\Phi} B$.

4º $A \delta_{\Phi} H$.

5º $B \delta_{\Phi} H$.

6º $B \delta_{\Phi} H$ (panel B not descriptively near panel H)

7º $H \delta_{\Phi} E$ (panel H not descriptively near panel E)

8º $A \delta_{\Phi} A$ (self-descriptive-proximity) ■

**Definition 3.** Soft Set. Let $X$ be a nonempty set, $A \in 2^{\mathbf{X}}$, and let $\varphi: 2^{\mathbf{X}} \to \mathbb{R}$ be defined by $|\varphi(A)| \le k \in \mathbb{Z}^{+}$, i.e., the length of the description of A is not definite. ■

**Definition 4.** Descriptively Proximal Soft Sets. Let $X$ be a physical nonempty set. $A, B \in 2^{\mathbf{X}}$ are descriptively proximal soft sets, provided

$$ \delta_{\text{lim}\Phi}(A, B) = \lim_{|\Phi(A) - \Phi(B)| \to 0} \delta_{\Phi}(A, B) = 0 $$

<figure><img src="image_3.png" /></figure>
**Remark 2.** The definition of $\delta_{lim\Phi}(A, B)$ implicitly permits the difference between the descriptions of A and B to vary without reaching identity, which would be the case for a pair of dynamical systems whose characteristics vary widely over a temporal interval [2]. That is, the value of $\delta_{lim\Phi}(A, B)$ is indefinite, which follows from Axiom 2. Hence, $\delta_{lim\Phi}(A, B)$ is termed a soft proximity. ■

**Definition 5. (Descriptive Proximity Space).**

For a nonempty set $X$, $(2^X \times 2^X, \delta_\Phi)$ is a descriptive proximity space. ■

**Definition 6. (Hausdorff Distance [3]).** Let $D(A, B)$ be the distance between sets of points A, B in the Euclidean plane $\mathbb{R}^2$. The Hausdorff distance $d_H(A, B)$ between $A, B \in 2^X$ is defined by

$$d_H(A, B) = \sup \{|D(x, A) - D(x, B)|, x \in \mathbb{R}^2\}$$

A Hausdorff metric space is denoted by $(2^X \times 2^X, d_H)$. ■

**Definition 7. (Descriptive Hausdorff Distance).** Let $D(\Phi(A), \Phi(B))$ be the distance between the descriptions of A, B in the Euclidean plane. The Descriptive Hausdorff distance $d_H(\Phi(A), \Phi(B))$ between the descriptions of A, B is defined by

$$d_H(\Phi(A), \Phi(B)) = \sup \{|D(x, \Phi(A)) - D(x, \Phi(B))|, x \in \mathbb{R}^2\}.$$

A descriptive Hausdorff metric space is denoted by $(2^{\Phi(X)}, d_H(\Phi(A), \Phi(B)))$. ■

**Definition 8. (Descriptive Hausdorff Topological Space).**

Let $A, B \in 2^X$. A descriptive Hausdorff topology $\tau_\Phi$ (DHT) on $(2^{\Phi(X)}, d_H(\Phi(A), \Phi(B)))$ has the following properties.

$$1^{\circ} \quad 2^{\Phi(X)}, \emptyset \in \tau_{\Phi}.$$

$$2^{\circ} \quad 2^{\Phi(A)} \cap 2^{\Phi(B)} \in \tau_{\Phi}.$$

$$3^{\circ} \quad 2^{\Phi(A)} \cup 2^{\Phi(B)} \in \tau_{\Phi}.$$

The triple $(2^{\Phi(X)}, d_H(\Phi(A), \Phi(B)), \tau_\Phi)$ is called a descriptive Hausdorff topological space. ■

**Lemma 1.** There is a descriptive Hausdorff topology $\tau_\Phi$ on every descriptive proximity space $(2^X \times 2^X, \delta_\Phi)$.

*Proof.* Replace $2^{\Phi(X)}$ (collection of descriptions) in Def. 8 with $2^{\delta_\Phi(X \times X)}$ (collection of descriptively proximal sets). Then we obtain a DHT on the descriptive proximity space. □

**Definition 9. (Descriptive Soft Proximity Space).**

For a nonempty set $X$, in place of $\delta_\Phi$, substitute $\delta_{lim\Phi}$ in Def. 5. Then $(2^X \times 2^X, \delta_{lim\Phi})$ is a descriptive soft proximity space. ■

**Theorem 1.** There is a descriptive Hausdorff topology on every descriptive Hausdorff proximity space.

*Proof.* Let $(2^X \times 2^X, \delta_{lim\Phi})$ denote a descriptive proximity space. We obtain the desired result from Def. 9 and Lemma 1, by observing that $(2^X \times 2^X, \delta_{lim\Phi})$ is
a descriptive proximity space, i.e., there is a DHT on the descriptive Hausdorff proximity space. □

**Remark 3.** The descriptive proximity space $(2^X \times 2^X, \delta_{lim\Phi})$ is considered soft, since $\delta_{lim\Phi}$ is defined on physical sets and Axiom 2 tells us that the description of every physical set is indefinite. ■

**Corollary 2.1.** There is a DHT on every descriptive soft proximity space $(2^X \times 2^X, \delta_{lim\Phi})$.

**Proof.** Immediate from Theorem 1. □

**Example 2.** Assume that $X$ is the collection of torus rectangular panels $A, B, C, E$ in Fig. 1 that map to flat counterparts $A, B, C, E$ in the Euclidean plane with descriptions $\varphi(H)$ that are areas

$$ \Phi(\mathbb{H}) := \int_{x_0}^{x_1} \mathbb{H}dx, \mathbb{H} \in \{A, B, C, E\}. $$

That is, $|\Phi(\mathbb{H})| = 1$. From Def. 3, each $\mathbb{H}$ is a soft set. Assume that each of the areas of the rectangles $A, B, C, E$ are equal. From Def. 4, each $H, H' \in 2^X$ are descriptively proximal soft sets, since

$$ \delta_{lim\Phi}(H, H') = \left| \int_{x_0}^{x_1} \mathbb{H}dx - \int_{x_0'}^{x_1'} \mathbb{H}'dx \right| = 0 \in \mathbb{R} $$

Hence, from Corollary 2.1, there is a DHT on the descriptive soft proximity space $(2^X \times 2^X, \delta_{lim\Phi})$. ■

### 3. COMPACTIFICATION OF A DESCRIPTIVE HAUSDORFF TOPOLOGICAL SPACE

This section introduces the compactification of a descriptive soft topological space.

**Definition 10. Open Ball** $B_d(p, \varepsilon)$ [6].

Consider a metric space $(X, d)$. The set

$$ B_d(p, \varepsilon) = \{x \in X : d(p, x) < \varepsilon\}, $$

with $\varepsilon > 0$, is called an open ball with centre $p \in X$ and radius $\varepsilon$. ■

**Definition 11. Open Set** $\mathring{A}$ [4].

A subset $A \subseteq X$ in the metric space $(X, d)$ is an open set (denoted by $\mathring{A}$), if and only if, for each point $x \in A$, there is an $\varepsilon > 0$ such that the open ball $B_d(x, \varepsilon) \subseteq A$. ■

**Definition 12. Open Cover** $\mathcal{F} := 2^{\{\mathring{A}\}}$.

A collection $\mathcal{F} := 2^{\{\mathring{A}\}}$ of open sets in a metric space $(X, d)$ is called an open cover of a subset $A \subset \mathcal{F}$, if and only if each point $x \in \mathring{A}$ belongs to $A$. ■

**Remark 4.** Let $\mathcal{F} := 2^{\{\mathring{A}\}}$ be an open cover of subset $A \subset \mathcal{F}$. Then a sub-collection $G \subset \mathcal{F}$ is a subcover of $A$, provided each point $x \in \mathring{A} \subset G$ belongs to $A$. ■
**Definition 13. Compact Metric Space.**

A subset $\mathbb{A}$ in a metric space $(X, d)$ is **compact**, if and only if every open cover of $\mathbb{A}$ has a finite subcover. If $\mathbb{A} = X$, then $(X, d)$ is termed a **compact metric space**.

**Remark 5.** Every descriptive Hausdorff metric is defined on a Euclidean space. It is also the case that every Euclidean space is non-compact. However, we also observe that every descriptive Hausdorff topological (DHT) space for a topology $\tau_{\Phi}$ on a subregion of a Euclidean space is locally compact, since every point in $\tau_{\Phi}$ has a compact neighborhood. ■

**Theorem 2.** [6].

A topological space is locally compact, provided each point of the space has a compact neighborhood.

**Definition 14. Descriptive Neighbourhood [6].**

Let $(2^X \times 2^X, \delta_\Phi)$ be a descriptive topological space, $A \in 2^X$, and $\Phi(A) \in \mathbb{R}^n$. A descriptive neighbourhood (denoted by $N_\Phi$) of $\Phi(A)$ in an open set $\mathring{A}$ such that $\Phi(A) \in \mathring{A}_{\delta_\Phi}$. An open set in a descriptive topology $\tau_\Phi$ is denoted by $[\mathring{A}_{\delta_\Phi}]$. ■

**Definition 15. Locally Compact Metric Space.**

A topological space is locally compact, if and only if each of the points in the space has a compact neighborhood. ■

**Theorem 3.** Every Descriptive Hausdorff topological space is locally compact.

*Proof.* Let $\tau_\Phi$ be a DHT on a descriptive Hausdorff proximity space $(2^X \times 2^X, \delta_\Phi)$ in a Hausdorff metric space $(2^X \times 2^X, d_H)$. Every descriptive proximity $\delta_\Phi(A, B) \in \tau_\Phi$ belongs to a descriptive neighborhood $N_\Phi(x)$, which is an open set $\mathring{A}_{\delta_\Phi}$. That is, each descriptive proximity $x \in N_\Phi(x)$ belongs to an open ball $B_{d_H}(x, \varepsilon) \subset N_\Phi$. Consequently, each of the points in $\tau_\Phi$ is in a compact neighborhood. Hence, from Def. 15, the $\tau_\Phi$ space is locally compact. □

**Theorem 4.** Every descriptively proximal soft topological space in a descriptive Hausdorff proximity space has a compactification.

*Proof.* Observe that descriptively proximal soft topology (DPST) in a descriptive Hausdorff proximity space is a DHT. Hence, from Theorem 3, the DPST is locally compact. □

### 4. APPLICATION: SATURN RING CHARGED WATER MOLECULES

In this section, we briefly introduce an application of the results we have given.

**Example 3.** Assume that the torus in Fig. 1 is one of the rings of Saturn and that the rectangular plates in the torus Fig. 1 are sets of particles (e.g., ring charged water molecules [5]) A, B, C, E, H swept along within a Saturn ring. Then observe 1º Each set of ring water molecules is a soft set (from Axiom 1).

2º Each pair of sets of ring water molecules are descriptively near sets (from Axiom 2). This observation makes it possible to organize the descriptions of ring water sets in a concise way.
3º Let $X$ be a set of ring water particles and let $2^X$ be a collection of subsets of ring particles. Let a descriptive Hausdorff proximity space $(2^X \times 2^X, \delta_\Phi)$ be equipped with the Hausdorff metric $d_H$. From Lemma 3, the descriptive Hausdorff topological space $\tau_\Phi$ on this proximity space is locally compact. This observation paves the way for the introduction of a variety of different open covers on $\tau_\Phi$, which would be a source of useful descriptions of Saturn ring water regions.

4º Observe that for every set $X$ of ring water molecules, the DHT space on the proximity space $((2^X \times 2^X, \tau_{lim\Phi}, d_H))$ is compactifiable (from Theorem 4).

5º From the result in Observation 4, one can put together one or more open covers on every DHT space on every ring water region descriptive proximity space. The beauty of such open covers is the ease with which one can tease from ring rain water observations, one or more precise open cover descriptions of each Saturn ring rain water region that contributes to the atmosphere of Saturn.

## 5. CONCLUSION

Compactification of a topological space is the long-standing emperor of topology. This paper introduces the emperor in new clothes, namely, in terms of descriptively proximal Hausdorff topological spaces. Recently, it has been found that there is a natural transition from proximal topologies to concise representations of complex physical systems (see, e.g., [6],[11, 13]).

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DEPARTMENT OF MATHEMATICS, FACULTY OF ARTS AND SCIENCES, IĞDIR UNIVERSITY, IĞDIR, TURKEY

*Email address: alkan.ozkan@igdir.edu.tr*

DEPARTMENT OF ELECTRICAL, UNIVERSITY OF MANITOBA, WINNIPEG, MB R3T 5V6 CANADA AND MATHEMATICS DEPARTMENT, ADIYAMAN UNIVERSITY, ADIYAMAN, TURKEY

*Email address: james.peters3@umanitoba.ca*

DEPARTMENT OF MATHEMATICS, FACULTY OF ARTS AND SCIENCES, IĞDIR UNIVERSITY, IĞDIR, 76000, TURKEY

*Email address: duman_ktu@hotmail.com*

DEPARTMENT OF APPLIED COMPUTER SCIENCE, UNIVERSITY OF WINNIPEG, WINNIPEG, MB R3B 2E9 CANADA

*Email address: s.ramanna@uwinnipeg.ca*