hcju/proofwikips huggingface dataset:
Definition:Population Parameter
A '''population parameter''' is a numerical description of a population.
Definition:Sample Statistic
A '''sample statistic''' is a numerical description of a sample.
Definition:Lagrange Basis Polynomial
Let $x_0, \ldots, x_n \in \R$ be real numbers. The '''Lagrange basis polynomials''' associated to the $x_i$ are the polynomials: :$\displaystyle \map {L_j} X := \prod_{\substack {0 \mathop \le i \mathop \le n \\ i \mathop \ne j} } \frac {X - x_i} {x_j - x_i} \in \R \sqbrk X$ {{explain|$\R \sqbrk X$}} {{mistake|Not sure if it's a mistake or a different way of defining it, but {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|entry = Lagrange interpolation formula}} has this wrapped up in another product symbol}} {{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}}
Definition:Final Topology
Let $X$ be a set. Let $I$ be an indexing set. Let $\family {\struct{Y_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $\family {f_i: Y_i \to X}_{i \mathop \in I}$ be an $I$-indexed family of mappings. === Definition 1 === {{:Definition:Final Topology/Definition 1}}
Definition:Convex Set (Order Theory)
=== Definition 1 === {{:Definition:Convex Set (Order Theory)/Definition 1}} === Definition 2 === {{:Definition:Convex Set (Order Theory)/Definition 2}}
Definition:Path-Connected/Metric Space
Let $M = \left({A, d}\right)$ be a metric space. $M$ is defined as '''path-connected''' iff: :$\forall m, n \in A: \exists f: \left[{0 \,.\,.\, 1}\right] \to A: f \left({0}\right) = m, f \left({1}\right) = n$ where $f$ is a continuous mapping. === Subset of Metric Space === {{:Definition:Path-Connected/Metric Space/Subset}}
Definition:Path-Connected/Metric Space/Subset
Let $M = \struct {A, d}$ be a metric space. Let $S \subseteq A$ be a subset of $M$. Then $S$ is '''path-connected (in $M$)''' : :$\forall m, n \in S: \exists f: \closedint 0 1 \to S: \map f 0 = m, \map f 1 = n$ where $f$ is a continuous mapping.
Definition:Percentile
Let $D$ be a finite collection of data regarding some quantitative variable. Let $D$ be divided into precisely $100$ classes. A $P$th '''percentile''' is a value in the interval defined by the $P$th class such that: :$\dfrac P {100}$ of the data in $D$ fall at or below the value chosen :$1 - \dfrac P {100}$ of the data in $D$ fall at or above the value chosen. Arbitrarily more precise definitions may be contrived as necessary to define a unique percentile for a given study.
Definition:Quartile
Let $D$ be a finite collection of data regarding some quantitative variable. Let $D$ be divided into precisely $4$ classes. A $Q$th '''quartile''' is a value in the interval defined by the $Q$th class such that: :$\dfrac Q 4$ of the data in $D$ fall at or below the value chosen; :$1 - \dfrac Q 4$ of the data in $D$ fall at or above the value chosen. Arbitrarily more precise definitions may be contrived as necessary to define a unique quartile for a given study. A common convention is: * The '''second quartile''', $Q_2$, is defined as the median of $D$ * The '''first quartile''', $Q_1$, is defined as the median of the data values below and not including $Q_2$  * The '''third quartile''', $Q_3$, is defined as the median of the data values above and not including $Q_2$
Definition:Interquartile Range
Let $Q_1$ and $Q_3$ be first and third quartiles. The '''interquartile range''' is defined and denoted as: :$\operatorname {IQR} := Q_3 - Q_1$
Definition:Coequalizer
Let $\mathbf C$ be a metacategory. Let $f, g: C \to D$ be morphisms with common domain and codomain. An '''equalizer''' for $f$ and $g$ is a morphism $q: D \to Q$ such that: :$q \circ f = q \circ g$ and subject to the following UMP: :For any $z: D \to Z$ such that $z \circ f = z \circ f$, there is a unique $u: Q \to Z$ such that: :::::$\begin{xy}\xymatrix{  C   \ar[r]<2pt>^*{f}   \ar[r]<-2pt>_*{g} &   D   \ar[r]^*{q}   \ar[rd]_*{z} &  Q   \ar@{.>}[d]^*{u} \\ & &  Z }\end{xy}$ :is a commutative diagram. I.e., $z = u \circ q$.
Definition:Ordering on Multiindices
Let $Z$ be the set of multiindices. We define an ordering on $Z$ as follows. If $k = \left \langle {k_j}\right \rangle_{j \in J}$ and $\ell = \left \langle {\ell_j}\right \rangle_{j \in J}$ are multiindices, then we write $k \preceq \ell$ if: :$\left( \forall j \in J \right) \left( k_j \le \ell_j \right)$ By abuse of notation, it is standard to use $\leq$ to denote the ordering on the set of multiindices as well.
Definition:Addition/Multiindices
The addition operation in the domain of multiindices $Z$ is written $+$. Let $k = \sequence {k_j}_{j \mathop \in J}$ and $l = \sequence {l_j}_{j \mathop \in J}$ be multiindices. Then $k + l$ is defined as: :$\paren {\forall j \in J} \paren {\paren {k + l}_j = k_j + l_j}$
Definition:Multiindex/Modulus
Let $k = \sequence {k_j}_{j \mathop \in J}$ be a multiindex. The '''modulus''' of such a multiindex $k$  is defined by: :$\displaystyle \size k = \sum_{j \mathop \in J} k_j$
Definition:Binomial Coefficient/Multiindices
Let $k = \left \langle {k_j}\right \rangle_{j \mathop \in J}$ and $\ell = \left \langle {\ell_j}\right \rangle_{j \mathop \in J}$ be multiindices. Let $\ell \le k$. Then $\dbinom k \ell$ is defined as: :$\displaystyle \binom k \ell = \prod_{j \mathop \in J} \binom {k_j} {\ell_j}$ Note that since by definition only finitely many of the $k_j$ are non-zero, the product in the definition of $\dbinom k \ell$ is convergent.
Definition:Power (Algebra)/Multiindices
Let $k = \left \langle {k_j}\right \rangle_{j = 1, \ldots, n}$ be a multiindex indexed by $\left\{{1, \ldots, n}\right\}$. Let $x = \left({x_1, \ldots, x_n}\right) \in \R^n$ be an ordered tuple of real numbers. Then $x^k$ is defined as: :$\displaystyle x^k := \prod_{j \mathop = 1}^n x_j^{k_j}$ where the powers on the right hand side are integer powers.
Definition:Coordinate System/Origin
The '''origin''' of a coordinate system is the zero vector. In the $x y$-plane, it is the point: :$O = \tuple {0, 0}$ and in general, in the Euclidean space $\R^n$: :$O = \underbrace {\tuple {0, 0, \ldots, 0} }_{\text{$n$ coordinates} }$ Thus it is the point where the axes cross over each other.
Definition:Neighborhood (Metric Space)
Let $M = \struct {A, d}$ be a metric space. Let $S \subseteq A$ be a subset of $A$. Let $x \in S$. Let there exist $\epsilon \in \R_{>0}$ such that the open $\epsilon$-ball at $x$ lies completely in $S$, that is: :$\map {B_\epsilon} x \subseteq S$ Then $S$ is a '''neighborhood of $x$ in $M$'''.
Definition:Opposite Magma
$\struct {S, *}$ is the '''opposite magma of $\struct {S, \circ}$''' : :$\forall x_1, x_2, x_3 \in S: x_1 \circ x_2 = x_3 \iff x_2 * x_1 = x_3$
Definition:Partial Differential Operator
Let $U \subseteq \R^n$ be a open set. Let $\CC \subseteq \map {\CC^k} {U, \R}$ be a set of $k$-times continuously differentiable functions. Let $\displaystyle \partial_i = \frac {\partial} {\partial x_i}$ denote the partial derivative, $i = 1, \ldots, n$. For a multiindex $\alpha = \tuple {\alpha_1, \ldots, \alpha_n}$ indexed by $\set {1, \ldots, n}$ let $\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}$. A mapping $T : \CC \to \map {\CC^k} {U, \R}$ is a '''partial differential operator''' if there exist $r \in \N$ and functions $f_\alpha : \R^n \to \C$ for each multiindex $\alpha$ with $\cmod \alpha \le r$ such that for all $g \in \CC$: :$\displaystyle \map T g = \sum_{\cmod \alpha \mathop \le r} f_\alpha \partial^\alpha g$
Definition:Region/Metric Space
Let $M = \left({A, d}\right)$ be a metric space. A '''region''' of $M$ is a subset $U$ of $M$ such that $U$ is: :$(1): \quad$ non-empty :$(2): \quad$ path-connected.
Definition:Open Set/Pseudometric Space
Let $P = \struct {A, d}$ be a pseudometric space. An '''open set''' in $P$ is defined in exactly the same way as for a metric space: $U$ is an '''open set in $P$''' : :$\forall y \in U: \exists \map \epsilon y > 0: \map {B_\epsilon} y \subseteq U$ where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.
Definition:Relatively Closed Set
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A \subseteq B \subseteq S$. === Definition 1 === {{:Definition:Relatively Closed Set/Definition 1}} === Definition 2 === {{:Definition:Relatively Closed Set/Definition 2}}
Definition:Closed Point
Let $a \in S$. Then '''$a$ is closed (in $T\,$)'''  $\left\{{a}\right\}$ is a closed set (in $T\,$).
Definition:Closed Set/Metric Space
{{:Definition:Closed Set/Metric Space/Definition 1}}
Definition:Bounded Mapping/Metric Space
Let $M$ be a metric space. Let $f: X \to M$ be a mapping from any set $X$ into $M$. Then $f$ is a '''bounded mapping'''  $\map f X$ is bounded in $M$.
Definition:Bounded Metric Space/Unbounded
Let $M = \left({X, d}\right)$ be a metric space. Let $M' = \left({Y, d_Y}\right)$ be a subspace of $M$. Then '''$M'$ is unbounded (in $M$)'''  $M'$ is not bounded in $M$.
Definition:Number Field
A '''number field''' is a subfield of the field of complex numbers.
Definition:Real Interval/Endpoints
The numbers $a, b \in \R$ are known as the '''endpoints''' of the interval. $a$ is sometimes called the '''left hand endpoint''' and $b$ the '''right hand endpoint''' of the interval.
Definition:Real Interval/Length
The difference $b - a$ between the endpoints is called the '''length''' of the interval.
Definition:Real Interval/Midpoint
The '''midpoint''' of a real interval is the number: : $\dfrac {a + b} 2$
Definition:Real Interval/Open
The '''open (real) interval''' from $a$ to $b$ is defined as: :$\openint a b := \set {x \in \R: a < x < b}$
Definition:Real Interval/Half-Open
There are two '''half-open (real) intervals''' from $a$ to $b$. ==== Right half-open ==== {{Definition:Real Interval/Half-Open/Right}} ==== Left half-open ==== {{Definition:Real Interval/Half-Open/Left}}
Definition:Real Interval/Closed
The '''closed (real) interval''' from $a$ to $b$ is defined as: :$\closedint a b = \set {x \in \R: a \le x \le b}$
Definition:Real Interval/Unbounded Closed
There are two '''unbounded closed intervals''' involving a real number $a \in \R$, defined as: {{begin-eqn}} {{eqn | l = \hointr a \to       | o = :=       | r = \set {x \in \R: a \le x} }} {{eqn | l = \hointl \gets a       | o = :=       | r = \set {x \in \R: x \le a} }} {{end-eqn}}
Definition:Real Interval/Unbounded Open
There are two '''unbounded open intervals''' involving a real number $a \in \R$, defined as: {{begin-eqn}} {{eqn | l = \openint a \to       | o = :=       | r = \set {x \in \R: a < x} }} {{eqn | l = \openint \gets a       | o = :=       | r = \set {x \in \R: x < a} }} {{end-eqn}} Using the same symbology, the set $\R$ can be represented as an '''unbounded open interval''' with no endpoints: :$\openint \gets \to = \R$
Definition:Real Interval/Empty
When $a > b$: {{begin-eqn}} {{eqn | l = \left [{a \,.\,.\, b} \right]       | m = \left\{ {x \in \R: a \le x \le b}\right\}       | mo = =       | r = \varnothing }} {{eqn | l = \left [{a \,.\,.\, b} \right)       | m = \left\{ {x \in \R: a \le x < b}\right\}       | mo = =       | r = \varnothing }} {{eqn | l = \left ({a \,.\,.\, b} \right]       | m = \left\{ {x \in \R: a < x \le b}\right\}       | mo = =       | r = \varnothing }} {{eqn | l = \left ({a \,.\,.\, b} \right)       | m = \left\{ {x \in \R: a < x < b}\right\}       | mo = =       | r = \varnothing }} {{end-eqn}} When $a = b$: : $\left [{a \,.\,.\, b} \right) = \left [{a \,.\,.\, a} \right) = \left\{{x \in \R: a \le x < a}\right\} = \varnothing$ : $\left ({a \,.\,.\, b} \right] = \left ({a \,.\,.\, a} \right] = \left\{{x \in \R: a < x \le a}\right\} = \varnothing$ : $\left ({a \,.\,.\, b} \right) = \left ({a \,.\,.\, a} \right) = \left\{{x \in \R: a < x < a}\right\} = \varnothing$ Such empty sets are referred to as '''empty intervals'''.
Definition:Real Interval/Singleton
When $a = b$: :$\closedint a b = \closedint a a = \set {x \in \R: a \le x \le a} = \set a$
Definition:Interval/Notation/Wirth
{{begin-eqn}} {{eqn | l = \openint a b       | o = :=       | r = \set {s \in S: \paren {a \prec s} \land \paren {s \prec b} }       | c = Open Interval }} {{eqn | l = \hointr a b       | o = :=       | r = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }       | c = Right Half-Open Interval }} {{eqn | l = \hointl a b       | o = :=       | r = \set {s \in S: \paren {a \prec s} \land \paren {s \preccurlyeq b} }       | c = Left Half-Open Interval }} {{eqn | l = \closedint a b       | o = :=       | r = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }       | c = Closed Interval }} {{end-eqn}}
Definition:Real Interval/Notation/Conventional
These are the notations usually seen for real intervals: {{begin-eqn}} {{eqn | l = \left ({a, b}\right)       | o = :=       | r = \set {x \in \R: a < x < b}       | c = Open real interval }} {{eqn | l = \left [{a, b}\right)       | o = :=       | r = \set {x \in \R: a \le x < b}       | c = Half-open real interval }} {{eqn | l = \left ({a, b}\right]       | o = :=       | r = \set {x \in \R: a < x \le b}       | c = Half-open real interval }} {{eqn | l = \left [{a, b}\right]       | o = :=       | r = \set {x \in \R: a \le x \le b}       | c = Closed real interval }} {{end-eqn}} but they can be confused with other usages for this notation. In particular, there exists the danger of taking $\paren {a, b}$ to mean an ordered pair.
Definition:Real Interval/Notation/Reverse-Bracket
In order to avoid the ambiguity problem arising from the conventional notation for intervals where an open real interval can be confused with an ordered pair, some authors use the '''reverse-bracket notation''' for open and half-open intervals: {{begin-eqn}} {{eqn | l = \left ] {\, a, b} \right [       | o = :=       | r = \set {x \in \R: a < x < b}       | c = Open real interval }} {{eqn | l = \left [ {a, b} \right [       | o = :=       | r = \set {x \in \R: a \le x < b}       | c = Half-open on the right }} {{eqn | l = \left ] {\, a, b} \right ]       | o = :=       | r = \set {x \in \R: a < x \le b}       | c = Half-open on the left }} {{end-eqn}} These are often considered to be both ugly ''and'' confusing, and hence are limited in popularity.
Definition:Interval/Notation/Unbounded Intervals
In Wirth interval notation, unbounded intervals of an ordered set $\struct {S, \preccurlyeq}$ are written as follows: {{begin-eqn}} {{eqn | l = \hointr a \to       | o = :=       | r = \set {x \in S: a \preccurlyeq x} }} {{eqn | l = \hointl \gets a       | o = :=       | r = \set {x \in S: x \preccurlyeq a} }} {{eqn | l = \openint a \to       | o = :=       | r = \set {x \in S: a \prec x} }} {{eqn | l = \openint \gets a       | o = :=       | r = \set {x \in S: x \prec a} }} {{eqn | l = \openint \gets \to       | o = :=       | r = \set {x \in S} = S }} {{end-eqn}}
Definition:Real Interval/Notation
An arbitrary (real) interval is frequently denoted $\mathbb I$. Sources which use the $\textbf {boldface}$ font for the number sets $\N, \Z, \Q, \R, \C$ tend also to use $\mathbf I$ for this entity. Some sources merely use the ordinary $\textit {italic}$ font $I$. === Wirth Interval Notation === {{:Definition:Real Interval/Notation/Wirth}}
Definition:Hereditarily Compact Space
Let $T = \left({S, \tau}\right)$ be a topological space. === Definition 1 === {{:Definition:Hereditarily Compact Space/Definition 1}} === Definition 2 === {{:Definition:Hereditarily Compact Space/Definition 2}}
Definition:Compact Space/Topology/Subspace
{{:Definition:Compact Space/Topology/Subspace/Definition 2}}
Definition:Compact Space/Euclidean Space
Let $\R^n$ denote Euclidean $n$-space. Let $H \subseteq \R^n$. Then $H$ is '''compact in $\R^n$'''  $H$ is closed and bounded. === Real Analysis === The same definition applies when $n = 1$, that is, for the real number line: {{:Definition:Compact Space/Real Analysis}} === Complex Analysis === {{:Definition:Compact Space/Complex Analysis}}
Definition:Subobject
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. A '''subobject''' of $C$ is a monomorphism $m: B \rightarrowtail C$ with codomain $C$.
Definition:Category of Subobjects
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. The '''category of subobjects of $C$''', denoted $\mathbf{Sub}_{\mathbf C} \left({C}\right)$, is defined as follows: {{DefineCategory |ob = Subobjects $m: B \to C$ of $C$ |mor = Morphisms $f: \operatorname{dom} m \to \operatorname{dom} m'$ of $\mathbf C$ such that $m' \circ f = m$ in $\mathbf C$ |comp = Inherited from $\mathbf C$ |id = $\operatorname{id}_m := \operatorname{id}_{\operatorname{dom} m}$, the identity morphism in $\mathbf C$ of the domain of $m$ }} The behaviour of the morphisms is shown in the following commutative diagram in $\mathbf C$: ::$\begin{xy}\xymatrix@+1em{  B   \ar[r]^*+{f}   \ar[rd]_*+{m} &  B'   \ar[d]^*+{m'} \\ &  C }\end{xy}$
Definition:Sphere/Geometry
A '''sphere''' is a surface in solid geometry such that all straight lines falling upon it from one particular point inside it are equal. {{EuclidDefinition|book = XI|def = 14|name = Sphere}}
Definition:Sphere/Geometry/Center
That point is called the '''center''' of the sphere.
Definition:Sphere/Geometry/Radius
A '''radius''' of a sphere is a straight line segment whose endpoints are the center and the surface of the sphere. '''The radius''' of a sphere is the length of one such radius.
Definition:Sphere/Topology
The $n$-dimensional '''sphere''', or '''$n$-sphere''', is the set: :$\Bbb S^n = \set {x \in \R^{n + 1} : \size {x - y} = r}$ where $\size {\, \cdot \, }$ denotes the Euclidean distance.
Definition:Sphere/Topology/Unit Sphere
The $n$-dimensional '''unit sphere''', or '''unit $n$-sphere''', is the $n$-sphere of radius $1$ and center the origin: :$\Bbb S^n = \left\{{x \in \R^{n+1} : \left\lvert{x}\right\rvert = 1}\right\}$
Definition:Inclusion Relation on Subobjects
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. The '''inclusion relation $\subseteq$''' on subobjects of $C$ is defined as follows: :$m \subseteq m'$ iff there exists a morphism $f: m \to m'$
Definition:Homotopy/Relative
Let $K \subseteq X$ be a subset of $X$. We say that $f$ and $g$ are '''homotopic relative to $K$''' if there exists a free homotopy $H$ between $f$ and $g$, and: :$(1): \quad \forall x \in K: f \left({x}\right) = g \left({x}\right)$ :$(2): \quad \forall x \in K, t \in \left[{0 \,.\,.\, 1}\right]: H \left({x, t}\right) = f \left({x}\right)$
Definition:Equivalent Subobjects
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. Two subobjects $m, m'$ of $C$ are said to be '''equivalent''' iff: :$m \subseteq m'$ and $m' \subseteq m$ where $\subseteq$ denotes the inclusion relation on subobjects.
Definition:Subobject Class
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. A '''subobject class of $C$''' is an equivalence class of subobjects of $C$ under the equivalence of subobjects. If $m$ is a subobject, its associated '''subobject class''' may be denoted by $\overline m$ or $\left[\!\left[{m}\right]\!\right]$. === Morphism Class === {{:Definition:Subobject Class/Morphism Class}}
Definition:Homotopy/Free
Let $X$ and $Y$ be topological spaces. Let $f: X \to Y$, $g: X \to Y$ be continuous mappings. Then $f$ and $g$ are '''(freely) homotopic'''  there exists a continuous mapping: : $H: X \times \left[{0 \,.\,.\, 1}\right] \to Y$ such that, for all $x \in X$: : $H \left({x, 0}\right) = f \left({x}\right)$ and: : $H \left({x, 1}\right) = g \left({x}\right)$ $H$ is called a '''(free) homotopy between $f$ and $g$'''.
Definition:Category of Subobject Classes
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$. The '''category of subobject classes of $C$''', denoted $\overline{\mathbf{Sub}}_{\mathbf C} \left({C}\right)$, is defined as follows: {{DefineCategory |ob = Subobject classes $\left[\!\left[{m}\right]\!\right]$ of $C$ |mor = Morphism classes $\left[\!\left[{f}\right]\!\right]: \left[\!\left[{m}\right]\!\right] \to \left[\!\left[{m'}\right]\!\right]$ |comp = Inherited from $\mathbf{Sub}_{\mathbf C} \left({C}\right)$, i.e. $\left[\!\left[{g}\right]\!\right] \circ \left[\!\left[{f}\right]\!\right] := \left[\!\left[{g \circ f}\right]\!\right]$ |id = $\operatorname{id}_{\left[\!\left[{m}\right]\!\right]} := \left[\!\left[{\operatorname{id}_m}\right]\!\right]$, the morphism class of the identity morphism of $m$ in $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ }}
Definition:Subobject Class/Morphism Class
Define the equivalence $\sim$ on the morphisms of $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ as follows. For morphisms $f: m \to n$ and $g: m' \to n'$ of $\mathbf{Sub}_{\mathbf C} \left({C}\right)$: :$f \sim g$ iff $m \sim m'$ and $n \sim n'$ where $m \sim m'$ signifies equivalence of subobjects. That $\sim$ in fact is an equivalence is shown on Morphism Class Equivalence is Equivalence. A '''morphism class''' is an equivalence class $\left[\!\left[{f}\right]\!\right]$ under $\sim$ of a morphism $f: m \to m'$. The domain and codomain of $\left[\!\left[{f}\right]\!\right]$ are taken to be $\left[\!\left[{m}\right]\!\right]$ and $\left[\!\left[{m'}\right]\!\right]$, respectively.
Definition:Inclusion Relation on Subobject Classes
Let $\mathbf C$ be a metacategory. Let $C$ be an object of $\mathbf C$. The '''inclusion relation $\subseteq$''' on subobject classes of $C$ is defined as follows: :$\left[\!\left[{m}\right]\!\right] \subseteq \left[\!\left[{m'}\right]\!\right]$ iff there exists a morphism $\left[\!\left[{f}\right]\!\right]: \left[\!\left[{m}\right]\!\right] \to \left[\!\left[{m'}\right]\!\right]$
Definition:Linear Continuum
A totally ordered set $\struct {S, \preceq}$ is a '''linear continuum''' : :$(1): \quad \struct {S, \preceq}$ is densely ordered :$(2): \quad \struct {S, \preceq}$ is Dedekind complete
Definition:Close Packed/Subset
A subset $T \subseteq S$ is said to be '''close packed in $\left({S, \preceq}\right)$''' : :$\forall a, b \in S: a \prec b \implies \exists c \in T: a \prec c \prec b$
Definition:Increasing/Sequence/Real Sequence
Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''increasing''' : :$\forall n \in \N: x_n \le x_{n + 1}$
Definition:Decreasing/Sequence/Real Sequence
Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''decreasing''' : :$\forall n \in \N: x_{n + 1} \le x_n$
Definition:Strictly Increasing/Sequence/Real Sequence
Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''strictly increasing''' : :$\forall n \in \N: x_n < x_{n + 1}$
Definition:Strictly Decreasing/Sequence/Real Sequence
Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''strictly decreasing''' : :$\forall n \in \N: x_{n + 1} < x_n$
Definition:Monotone (Order Theory)/Sequence/Real Sequence
Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''monotone'''  it is either increasing or decreasing.
Definition:Strictly Monotone/Sequence/Real Sequence
Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is '''strictly monotone'''  it is either strictly increasing or strictly decreasing.
Definition:Archimedean Property/Norm
Let $n: S \to \R$ be a norm on $S$. {{explain|What is a norm on a general algebraic structure?}} {{Disambiguate|Definition:Norm}} Then $n$ satisfies the '''Archimedean property on $S$''' : :$\forall a, b \in S: n \paren a < n \paren b \implies \exists m \in \N: n \paren {m \cdot a} > n \paren b$ Using the more common symbology for a norm: :$\forall a, b \in S: \norm a < \norm b \implies \exists m \in \Z_{>0}: \norm {m \cdot a} > \norm b$
Definition:Archimedean Property/Ordering
Let $\preceq$ be an ordering on $S$. Then $\preceq$ satisfies the '''Archimedean property on $S$''' : :$\forall a, b \in S: a \prec b \implies \exists m \in \Z_{>0}: b \prec m \cdot a$