hcju/proofwikips huggingface dataset:
Definition:Relatively Prime Integer Topology
Let $\Z_{>0}$ denote the set of (strictly) positive integers. Let $\BB$ be the set of sets $\set {\map {U_a} b: a, b \in \Z_{>0} }$ where: :$\map {U_a} b = \set {b + n a \in \Z_{>0}: \gcd \set {a, b} = 1}$ where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$. Then $\BB$ is the basis for a topology $\tau$ on $\Z_{>0}$. $\tau$ is then referred to as the '''relatively prime integer topology'''. The topological space $T = \struct {\Z_{>0}, \tau}$ is referred to as the '''relatively prime integer space'''.
Definition:Smirnov's Deleted Sequence Topology
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $A$ denote the set defined as: :$A := \set {\dfrac 1 n: n \in \Z_{>0} }$ Let $\tau$ be the topology defined as: :$\tau = \set {H: \exists U \in \tau_d, B \subseteq A: H = U \setminus B}$ That is, $\tau$ consists of the open sets of $\struct {\R, \tau_d}$ which have had any number of the set of the reciprocals of the positive integers removed. $\tau$ is then referred to as '''Smirnov's deleted sequence topology on $\R$'''.
Definition:Indiscrete Extension of Reals
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $D$ be an everywhere dense subset of $\struct {\R, \tau_d}$ with an everywhere dense complement in $\R$. Let $\BB$ be the set of sets: :$\BB := \set {H: \exists U \in \tau_d: H = U \cap D}$ Let $\tau^*$ be the topology generated from $\tau_d$ by the addition of all sets of $\BB$. :$\tau^* = \tau_d \cup \BB$ $\tau^*$ is then referred to as an '''indiscrete extension of $\R$'''.
Definition:Pointed Extension of Reals
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $D$ be an everywhere dense subset of $\struct {\R, \tau_d}$ with an everywhere dense complement in $\R$. Let $\BB$ be the set of sets defined as: :$\BB = \set {\set x \cup \paren {U \cap D}: x \in U \in \tau_d}$ Let $\tau'$ be the topology generated from $\BB$. $\tau'$ is referred to as a '''pointed extension of $\R$'''.
Definition:Indiscrete Extension of Reals/Rational
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Q$ denote the set of rational numbers. Let $\tau^*$ be the indiscrete extension of $\struct {\R, \tau_d}$: :$\tau^* = \tau_d \cup \set {H: \exists U \in \tau_d: H = U \cap \Q}$ $\tau^*$ is then referred to as the '''indiscrete rational extension of $\R$'''.
Definition:Pointed Extension of Reals/Irrational
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers. Let $\BB$ be the set of sets defined as: :$\BB = \set {\set x \cup \paren {U \cap \Bbb I}: x \in U \in \tau_d}$ Let $\tau'$ be the topology generated from $\BB$. $\tau'$ is referred to as '''pointed irrational extension of $\R$'''.
Definition:Pointed Extension of Reals/Rational
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Q$ denote the set of rational numbers. Let $\BB$ be the set of sets defined as: :$\BB = \set {\set x \cup \paren {U \cap \Q}: x \in U \in \tau_d}$ Let $\tau'$ be the topology generated from $\BB$. $\tau'$ is referred to as the '''pointed rational extension of $\R$'''.
Definition:Discrete Extension of Reals
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $D$ be an everywhere dense subset of $\struct {\R, \tau_d}$ with an everywhere dense complement in $\R$. Let $\BB$ be the set of sets defined as: :$\BB = \tau_d \cup \set {\set x: x \in D}$ Let $\tau*$ be the topology generated from $\BB$. $\tau^*$ is referred to as a '''discrete extension of $\R$'''.
Definition:Discrete Extension of Reals/Rational
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Q$ denote the set of rational numbers. Let $\BB$ be the set of sets defined as: :$\BB = \tau_d \cup \set {\set x: x \in \Q}$ Let $\tau*$ be the topology generated from $\BB$. $\tau^*$ is referred to as the '''discrete rational extension of $\R$'''.
Definition:Discrete Extension of Reals/Irrational
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers. Let $\BB$ be the set of sets defined as: :$\BB = \tau_d \cup \set {\set x: x \in \Bbb I}$ Let $\tau*$ be the topology generated from $\BB$. $\tau^*$ is referred to as the '''discrete irrational extension of $\R$'''.
Definition:Telophase Topology
Let $S = \closedint 0 1 \cup \set {1^*}$ where: :$\closedint 0 1$ is the closed unit interval $\set {x \in \R: 0 \le x \le 1}$ :$1^*$ is a second right hand endpoint of $\closedint 0 1$. Let $\BB$ be a local basis defined as: :$\BB = \set {\openint a 1 \cup \set {1^*}: a \in \closedint 0 1}$ Let $\tau$ be the topology generated from $\BB$. $\tau$ is referred to as the '''telophase topology'''.
Definition:Simple Matroid
Let $M = \struct{S, \mathscr I}$ be a matroid. $M$ is called a '''simple matroid''' if it has no loops or parallel elements.
Definition:Irrational Slope Topology
Let $S = \set {\tuple {x, y}: y \ge 0, x, y \in \Q}$. Let $\theta \in \R \setminus \Q$ be some fixed irrational number. :500pxrightthumb$\epsilon$-neighborhood $\map {N_\epsilon} {x, y}$: intervals are of [[Definition:Length of Real Intervallength $\epsilon$]] Let $\BB$ be the set of $\epsilon$-neighborhoods of the form: :$\map {N_\epsilon} {x, y} = \set {\tuple {x, y} } \cup \map {B_\epsilon} {x + \dfrac y \theta} \cup \map {B_\epsilon} {x - \dfrac y \theta}$ where: :$\map {B_\epsilon} \zeta := \set {r \in \Q: \size {r - \zeta} < \epsilon}$ that is, the open $\epsilon$-ball at $\zeta$ in $\Q^2$. Hence each $\map {N_\epsilon} {x, y}$ consists of: :the singleton $\set {\tuple {x, y} }$ together with: :two open intervals of length $\epsilon$ in the rational numbers whose midpoints are at the irrational points $x \pm \dfrac y \theta$ The lines joining $\tuple {x, y}$ to $\tuple {x \pm \dfrac y \theta, 0}$ have slope $\pm \theta$ and are not part of $\map {N_\epsilon} {x, y}$. Let $\tau$ be the topology generated from $\BB$. $\tau$ is referred to as the '''irrational slope topology'''.
Definition:Deleted Diameter Topology
Let $\struct {\R^2, \tau_d}$ be the real number plane with the usual (Euclidean) topology induced by the Euclidean metric $d$. Let $\BB$ be the sub-basis of $\tau_d$ defined as the set of all open balls of $\struct {\R^2, d}$ with the horizontal diameters apart from the centers excluded. Let $\sigma$ be the topology generated from $\BB$. $\sigma$ is referred to as the '''deleted diameter topology'''.
Definition:Deleted Radius Topology
Let $\struct {\R^2, \tau_d}$ be the real number plane with the usual (Euclidean) topology induced by the Euclidean metric $d$. Let $\BB$ be the sub-basis of $\tau_d$ defined as the set of all open balls of $\struct {\R^2, d}$ with the horizontal radii to the right of the centers excluded. Let $\tau$ be the topology generated from $\BB$. $\tau$ is referred to as the '''deleted radius topology'''.
Definition:Half-Plane/Left
=== Open Left Half-Plane === {{:Definition:Half-Plane/Open/Left}} === Closed Left Half-Plane === {{:Definition:Half-Plane/Closed/Left}}
Definition:Half-Plane/Right
=== Open Right Half-Plane === {{:Definition:Half-Plane/Open/Right}} === Closed Right Half-Plane === {{:Definition:Half-Plane/Closed/Right}}
Definition:Half-Plane/Upper
=== Open Upper Half-Plane === {{:Definition:Half-Plane/Open/Upper}} === Closed Upper Half-Plane === {{:Definition:Half-Plane/Closed/Upper}}
Definition:Half-Plane/Lower
=== Open Lower Half-Plane === {{:Definition:Half-Plane/Open/Lower}} === Closed Lower Half-Plane === {{:Definition:Half-Plane/Closed/Lower}}
Definition:Half-Plane/Open/Left
The '''open left half-plane''' $H_{\text {OL} }$ is the area of $P$ on the left of $L$. That is, where $x < 0$: :$H_{\text {OL} } := \set {\tuple {x, y}: x \in \R_{<0} }$
Definition:Half-Plane/Closed/Left
The '''closed left half-plane''' $H_{\text {CL} }$ is the area of $P$ on the left of and including $L$. That is, where $x \le 0$: :$H_{\text {CL} } := \set {\tuple {x, y}: x \in \R_{\le 0} }$
Definition:Half-Plane/Closed/Right
The '''closed right half-plane''' $H_{\text {CR} }$ is the area of $P$ on the right of and including $L$. That is, where $x \ge 0$: :$H_{\text {CR} } := \set {\tuple {x, y}: x \in \R_{\ge 0} }$
Definition:Half-Plane/Closed/Upper
The '''closed upper half-plane''' $H_{\text {CU} }$ is the area of $P$ above and including $L$. That is, where $y \ge 0$: :$H_{\text {CU} } := \set {\tuple {x, y}: y \in \R_{\ge 0} }$
Definition:Half-Plane/Closed/Lower
The '''closed lower half-plane''' $H_{\text {CL} }$ is the area of $P$ below and including $L$. That is, where $y \le 0$: :$H_{\text {CL} } := \set {\tuple {x, y}: y \in \R_{\le 0} }$
Definition:Half-Plane/Open/Lower
The '''open lower half-plane''' $H_{\text {OL} }$ is the area of $P$ below $L$. That is, where $y < 0$: :$H_{\text {OL} } := \set {\tuple {x, y}: y \in \R_{< 0} }$
Definition:Half-Plane/Open/Upper
The '''open upper half-plane''' $H_{\text {OU} }$ is the area of $P$ above $L$. That is, where $y > 0$: :$H_{\text {OU} } := \set {\tuple {x, y}: y \in \R_{> 0} }$
Definition:Half-Plane/Open/Right
The '''open right half-plane''' $H_{\text {OR} }$ is the area of $P$ on the right of $L$. That is, where $x > 0$: :$H_{\text {OR} } := \set {\tuple {x, y}: x \in \R_{>0} }$
Definition:Half-Disc Topology
Let $P = \set {\tuple {x, y}: x \in \R, y \in \R_{>0} }$ be the open upper half-plane.  Let $\struct {P, \tau_d}$ be the open upper half-plane with the Euclidean topology. Let $L$ denote the $x$-axis Let $\BB$ be the set of sets of the form: :$\set x \cup \paren {U \cap P}$ where: :$x \in L$ :$U$ is a Euclidean neighborhood of $x$. Let $\tau^*$ be the topology generated from $\BB$. $\tau^*$ is referred to as the '''half-disc topology'''.
Definition:Irregular Lattice Topology
Let $S$ be the subset of the lattice points of the Cartesian plane defined as: :$S := \set {\tuple {i, j}: i, j \in \Z_{>0} } \cup \set {\tuple {i, 0}: i \in \Z_{\ge 0} }$ Let each point of $S$ be defined as being open. Let each point of $S$ of the form $\tuple {i, 0}$ such that $i \ne 0$ have as a local basis sets $U_n$ of the form: :$\map {U_n} {i, 0} := \set {\tuple {i, k}: k = 0 \text { or } k \ge n}$ Let the point $\tuple {0, 0}$ have as a local basis a set $V_n$ of the form: :$V_n := \set {\tuple {i, k}: i = k = 0 \text { or } i, k \ge n}$ Let $\tau$ be the topology generated from all these $U_n$ and $V_n$. $\tau$ is referred to as the '''irregular lattice topology'''.
Definition:Lattice Point/Rational
$P$ is a '''rational lattice point''' of $\CC$  $a_1, a_2, \ldots, a_n$ are all rational numbers.
Definition:Arens Square
Let $A$ be the subset of the unit square defined as: :$A := \set {\tuple {i, j}: 0 < i < 1, 0 < j < 1, i, j \in \Q} \setminus \set {\tuple {i, j}: i = \dfrac 1 2}$ That is: :$A := \openint 0 1^2 \cup \Q \times \Q \setminus \set {\tuple {\dfrac 1 2}, j: j \in \Q}$ That is, the set of rational points in the interior of the unit square from which are excluded the points whose $x$ coordinates are equal to $\dfrac 1 2$. :rightthumb300pxArens square with example local bases marked Let $S$ be the set defined as: :$S = A \cup \set {\tuple {0, 0} } \cup \set {\tuple {1, 0} } \cup \set {\tuple {\dfrac 1 2, r \sqrt 2}: r \in \Q, 0 < r \sqrt 2 < 1}$ Let $\BB$ be the basis for a topology generated on $S$ be defined by granting: :to each point of $A$ the local basis of open sets inherited by $A$ from the Euclidean topology on the unit square; :to the other points of $S$ the following local bases: {{begin-eqn}} {{eqn | l = \map {U_n} {0, 0}       | o = :=       | r = \set {\tuple {x, y}: 0 < x < \dfrac 1 4, 0 < y < \dfrac 1 n} \cup \set {\tuple {0, 0} } }} {{eqn | l = \map {U_n} {1, 0}       | o = :=       | r = \set {\tuple {x, y}: \dfrac 3 4 < x < 1, 0 < y < \dfrac 1 n} \cup \set {\tuple {1, 0} } }} {{eqn | l = \map {U_n} {\tuple {\dfrac 1 2, r \sqrt 2} }       | o = :=       | r = \set {\tuple {x, y}: \dfrac 1 4 < x < \dfrac 3 4, \size {y - r \sqrt 2} < \dfrac 1 n} }} {{end-eqn}} Let $\tau$ be the topology generated from $\BB$. $\struct {S, \tau}$ is referred to as the '''Arens square'''.
Definition:Simplified Arens Square
Let $A$ be the set of points in the interior of the unit square: :$A := \set {\tuple {i, j}: 0 < i < 1, 0 < j < 1, i, j \in \R} = \openint 0 1^2$ :rightthumb300pxSimplified Arens square with example local bases marked Let $S$ be the set defined as: :$S = A \cup \set {\tuple {0, 0} } \cup \set {\tuple {1, 0} }$ Let $\BB$ be the basis for a topology generated on $S$ be defined by granting: :to each point of $A$ the local basis of open sets inherited by $A$ from the Euclidean topology on the unit square; :to the other points of $S$ the following local bases: {{begin-eqn}} {{eqn | l = \map {U_n} {0, 0}       | o = :=       | r = \set {\tuple {x, y}: 0 < x < \dfrac 1 2, 0 < y < \dfrac 1 n} \cup \set {\tuple {0, 0} } }} {{eqn | l = \map {U_m} {1, 0}       | o = :=       | r = \set {\tuple {x, y}: \dfrac 1 2 < x < 1, 0 < y < \dfrac 1 m} \cup \set {\tuple {1, 0} } }} {{end-eqn}} Let $\tau$ be the topology generated from $\BB$. $\struct {S, \tau}$ is referred to as the '''simplified Arens square'''.
Definition:Metrizable Tangent Disc Topology
Let $A$ be a countable subset of the $x$-axis in the Cartesian plane $\R^2$. Let $P$ denote the open upper half-plane in $\R^2$. Let $S := A \cup P$. Let $\struct {S, \tau}$ be the topological subspace of the Niemytzki plane. $\struct {S, \tau}$ is referred to as the '''metrizable tangent disc topology'''.
Definition:Sorgenfrey's Half-Open Square Topology
Let $A = \struct {\R, \sigma}$ be a Sorgenfrey line. Let $S = A \times A$ denote the product space of $A$ with itself, whose topology $\tau$ is such that: :the neighborhood of a point $\tuple {p, q}$ is a rectangle in $\R^2$ of the form: ::$\set {\set {x, y}: p \le x < p + \epsilon_1, q \le y < q + \epsilon_2}$ :for some $\epsilon_1, \epsilon_2 \in \R_{>0}$. $\struct {S, \tau}$ is referred to as '''Sorgenfrey's half-open square topology'''.
Definition:Michael's Product Topology
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers. Let $\struct {S, \sigma} := \struct {\R, \tau^*} \times \struct {\Bbb I, \tau'}$, where: :$\tau^*$ is the discrete irrational extension of $\tau_d$ by $\Bbb I$ :$\tau'$ is the subspace topology on $\Bbb I$ induced by $\tau_d$. $\struct {S, \sigma}$ is referred to as '''Michael's product topology'''.
Definition:Tychonoff Plank
Let $\omega$ be the first transfinite ordinal. Let $\Omega$ be the first uncountable ordinal. Let $\closedint 0 \Omega$ and $\closedint 0 \omega$ be closed ordinal spaces which have both been given the interval topology. The '''Tychonoff plank''' is the topological space defined as: :$T = \closedint 0 \Omega \times \closedint 0 \omega$
Definition:Tychonoff Plank/Deleted
Let $S = \closedint 0 \Omega$ and $\closedint 0 \omega$ be closed ordinal spaces which have both been given the interval topology. Hence let $T = \struct {S, \tau}$ denote the Tychonoff plank. The '''deleted Tychonoff plank''' is the topological subspace defined as: :$T_\infty = \struct {S \setminus \set {\tuple {\Omega, \omega} }, \tau}$
Definition:Alexandroff Plank
:rightthumb600px Let $\Omega$ be the first uncountable ordinal. Let the closed ordinal space $\closedint 0 \Omega$ be given the interval topology. Let the closed real interval $\closedint {-1} 1$ be given the interval topology. Let $\struct {S, \tau}$ be the product space of $\closedint 0 \Omega$ with $\closedint {-1} 1$ Consider the point $p := \tuple {\Omega, 0} \in S$ Let $\sigma$ be the expansion of $\tau$ generated by adding to $\tau$ the sets of the form: :$\map U {\alpha, n} := \set p \cup \hointl \alpha \Omega \times \openint 0 {\dfrac 1 n}$ Then the topological space $T = \struct {S, \sigma}$ is known as the '''Alexandroff plank'''.
Definition:Dieudonné Plank
:rightthumb600px Let $\omega$ be the first transfinite ordinal. Let $\Omega$ be the first uncountable ordinal. Let $S = \paren {\closedint 0 \Omega \times \closedint 0 \omega} \setminus \set {\tuple {\Omega, \omega} }$ be the underlying set of the deleted Tychonoff plank. Let the topology $\tau$ be generated by declaring as open: :each point of $\hointr 0 \Omega \times \hointr 0 \omega$ together with the sets defined as: {{begin-eqn}} {{eqn | l = \map {U_\alpha} \beta       | r = \set {\tuple {\beta, \gamma}: \alpha < \gamma \le \omega} }} {{eqn | l = \map {V_\alpha} \beta       | r = \set {\tuple {\gamma, \beta}: \alpha < \gamma \le \Omega} }} {{end-eqn}} The topological space $\struct {S, \tau}$ is known as the '''Dieudonné plank'''.
Definition:Tychonoff Corkscrew
Let $\omega$ be the first transfinite ordinal. Let $\Omega$ be the first uncountable ordinal. For each ordinal $\alpha$, let $A_\alpha$ denote the totally ordered set defined as: :$A_\alpha := \set {-0, -1, -2, \ldots, \alpha, \ldots, 2, 1, 0}$ with the order topology. {{explain|What is the meaning of the above ordering?}} :thumbright600pxOne instance of the plane $P^*$ Let $P$ be the product space defined as: :$P := A_\Omega \times A_\omega$ Let $P^*$ be the topological subspace of $P$: :$P^* := P \setminus \set {\tuple {\Omega, \omega} }$ Consider an infinite stack of copies of $P^*$. Let a rectangular corkscrew lattice $C$ be formed such that it spirals in both directions by: :$(1): \quad$ cutting each $P^*$ immediately below the positive $A_\Omega$ axis $\tuple {\Omega, \omega}$ to $\tuple {0, \omega}$ :$(2): \quad$ joining the fourth quadrant of one $P^*$ to the first quadrant of the $P^*$ immediately below it, along the positive $A_\Omega$ axis. Let $\sequence {\map { {A_\Omega}^+} i}_{i \mathop = -\infty}^{+ \infty}$ be the family of positive $A_\Omega$ axes. Then $i$ will be referred to as the '''level''' of $A_\Omega$. By convention, the points of $C$ which lie above $\map { {A_\Omega}^+} i$ are at a level '''greater''' than $i$. If $x$ is such a point, this is denoted $\map L x > i$. We create $S$ by adding to $C$ two distinguished points $a^+$ and $a^-$ which can be considered as being infinity points at the top and bottom of the axis of the corkscrew. Basis neighborhoods of $a^+$ consist of all points of $S$ which lie above a certain level. Basis neighborhoods of $a^-$ consist of all points of $S$ which lie below a certain level. The topological space $\struct {S, \tau}$ so generated is referred to as the '''Tychonoff corkscrew'''.
Definition:Tychonoff Corkscrew/Deleted
Let $T = \struct {S, \tau}$ denote the Tychonoff corkscrew. Let $a^-$ denote the distinguished point of $S$ considered as being the infinity point at the bottom of $T$. The '''deleted Tychonoff corkscrew''' is the topological subspace defined as: :$T_\infty = \struct {S \setminus \set {a^-}, \tau}$
Definition:Thomas's Plank/Definition 1
Let $L_n$ be lines embedded in the Cartesian plane $\R^2$ defined as: :$\forall n \in \N: L_n = \begin {cases} \set {\tuple {x, 0}: x \in \openint 0 1} & : n = 0 \\ \set {\tuple {x, \dfrac 1 n}: x \in \hointr 0 1} & : n > 0 \end {cases}$ Let $S = \displaystyle \bigcup {n \mathop \in \N}$. Let a topology $\tau$ be applied to $S$, defined as follows: :For $n \ge 1$, each point of $L_n$ except for $\tuple {0, \dfrac 1 n}$ is open. :neighborhood bases of $\tuple {0, \dfrac 1 n}$ are subsets of $L_n$ with finite complements. :neighborhood bases of $\tuple {x, 0}$ are the sets $\map {U_i} {x, 0}$ defined as: ::$\map {U_i} {x, 0} := \set {\tuple{x, \dfrac 1 n}: n > i}$ '''Thomas's plank''' is the topological space $\struct {S, \tau}$.
Definition:Thomas's Plank/Definition 2
Let $L := \openint 0 1$ denote the open unit interval. Let $S_1 := L \cup \set p$ denote the Alexandroff extension of $L$. Let $S_2 := \Z_{>0} \cup \set q$ denote the Alexandroff extension of the (strictly) positive integers $\Z_{>0}$. Let $\struct {S, \tau} := \paren {S_1 \times S_2} \setminus \set {\tuple {p, q} }$ be the subspace of the product space $S_1 \times S_2$ with $\set {\tuple {p, q} }$ removed. '''Thomas's plank''' is the topological space $\struct {S, \tau}$.
Definition:Thomas's Corkscrew
Let copies of Thomas's plank be used to construct a corkscrew in the same manner as the Tychonoff plank is used to construct the Tychonoff corkscrew. '''Thomas's corkscrew''' is the topological space that results.
Definition:Parallel Line Topology/Strong
Let $\BB$ be the set of sets of the form: {{begin-eqn}} {{eqn | l = \map V {a, b}       | r = \set {\paren {x, 1}: a \le x < b} }} {{eqn | l = \map U {a, b}       | r = \set {\paren {x, 0}: a < x \le b} \cup \set {\paren {x, 1}: a < x \le b} }} {{end-eqn}} that is: :the left half-open real intervals on $B$ and: :the right half-open real intervals on $A$ together with the interior of their projection onto $B$. $\BB$ is then taken to be the basis for a topology $\sigma$ on $S$. Thus $\sigma$ is referred to as the '''strong parallel line topology'''. The topological space $T = \struct {S, \sigma}$ is referred to as the '''strong parallel line space'''.
Definition:Parallel Line Topology/Weak
Let $\BB$ be the set of sets of the form: {{begin-eqn}} {{eqn | l = \map U {a, b}       | r = \set {\paren {x, 0}: a < x \le b} \cup \set {\paren {x, 1}: a < x \le b} }} {{eqn | l = \map W {a, b}       | r = \set {\paren {x, 0}: a < x < b} \cup \set {\paren {x, 1}: a \le x < b} }} {{end-eqn}} that is: :the left half-open real intervals on $B$ together with the interior of their projection onto $A$ and: :the right half-open real intervals on $A$ together with the interior of their projection onto $B$. $\BB$ is then taken to be the basis for a topology $\tau$ on $S$. Thus $\tau$ is referred to as the '''weak parallel line topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''weak parallel line space'''.
Definition:Concentric Circle Topology
:rightthumb280pxOpen Set of $T$ Let $C_1$ and $C_2$ be concentric circles in the Cartesian plane $\R^2$ such that $C_1$ is inside $C_2$. Let $S = C_1 \cup C_2$. Let $\BB$ be the set of sets consisting of: :all singleton sets of $C_2$ :all open intervals on $C_1$ each together with its projection from the center of the circles onto $C_2$ except for the midpoint. $\BB$ is then taken to be the sub-basis for a topology $\tau$ on $S$. Thus $\tau$ is referred to as the '''concentric circle topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''concentric circle space'''.
Definition:Appert Space
Let $\Z_{>0}$ denote the (strictly) positive integers. For a given subset $H$ of $\Z_{>0}$ and element $n$ of $\Z_{>0}$, let $\map N {n, H}$ denote the number of integers in $E$ which are less than or equal to $n$: :$\forall n \in \Z_{>0}, H \subseteq \Z_{>0}: \map N {n, H} = \card {\set {x \in H: x \le n} }$ Let $\tau$ be the topology defined as: :$1 \notin H \implies H \in \tau$ :$1 \in H$ and $\displaystyle \lim_{n \mathop \to \infty} \dfrac {\map N {n, H} } n = 1 \implies H \in \tau$ for $H \subseteq \Z_{>0}$. $\tau$ is referred to as the '''Appert topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''Appert space'''.
Definition:Maximal Compact Topology
Let $X = \Z_{>0} \times \Z_{>0}$ be the set of all lattice points $\tuple {i, j}$ of the Cartesian plane where $i$ and $j$ are both (strictly) positive integers. Let $S = X \cup \set {x, y}$ where $x$ and $y$ are two new elements of $X$. Let $\tau$ be the topology defined on $S$ as follows: :Each lattice point of $S$ is an open point. :The open neighborhoods of $x$ are of the form $S \setminus A$ where $A$ is any set of lattice points with at most finitely many points on each row :The open neighborhoods of $y$ are of the form $S \setminus B$ where $B$ is any set of lattice points selected from at most finitely many rows. $\tau$ is referred to as the '''maximal compact topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''maximal compact space'''.