List of Topological Spaces by Separation and Compactness

Introduction

This article lists the separation axioms and the compactness conditions that Parts I to III of the corpus use, and tabulates which of the spaces of the corpus satisfies which. The separation axioms record how far a space separates points from points, points from closed sets, and closed sets from closed sets; the compactness conditions record how far the open covers of a space can be reduced to a finite, a countable or a locally finite family. The two families are the ones that decide which theorems apply to a space, and the article collects them in one place so that the same space can be read down a column.

Every entry points to the article that introduces the space, the axiom or the condition, and every space tabulated appears in Parts I to III. The article introduces nothing and proves nothing: it records the implications that the corpus establishes and the examples that show the implications do not reverse, and it does not restate a definition or give a proof.

The article records examples and non-examples side by side. Beside the spaces that satisfy an axiom or a compactness condition it lists the spaces that fail it — the cofinite topology on an infinite set, which is $T_1$ and not Hausdorff; the Sorgenfrey plane, which is Tychonoff and not normal; Mysior's example, which is regular and not completely regular; the long line, which is countably compact and not compact and not paracompact — each with the failure named and the article that records it.

The Separation Axioms

The axioms are those of Topological Spaces and Metrisation and Separation Axioms, where the hierarchy, its permanence and the metrisation theorems are proved. The convention of the corpus is that $T_3$, $T_{3\frac12}$ and $T_4$ denote the corresponding condition together with $T_1$.

Axiom	What it separates	Introduced in
$T_0$	any two distinct points, by an open set containing exactly one of them	Topological Spaces
$T_1$	any two distinct points, by open sets each avoiding the other; equivalently points are closed	Topological Spaces
Hausdorff ($T_2$)	any two distinct points, by disjoint open sets; equivalently limits of nets are unique	Topological Spaces
regular ($T_3$)	a point from a closed set not containing it, by disjoint open sets; with $T_1$, the regular Hausdorff spaces	Metrisation and Separation Axioms
completely regular ($T_{3\frac12}$)	a point from a closed set not containing it, by a continuous function to $[0,1]$; with $T_1$, the Tychonoff spaces	Metrisation and Separation Axioms
normal ($T_4$)	two disjoint closed sets, by disjoint open sets; with $T_1$, the normal Hausdorff spaces	Metrisation and Separation Axioms
perfectly normal	two disjoint closed sets, by a continuous function; every closed set a countable intersection of open sets	Metrisation and Separation Axioms
collectionwise normal	a discrete family of closed sets, by a discrete family of open sets	Paracompactness and Partitions of Unity

The implications run one way only:

Implication	The condition that strengthens it	Introduced in
$T_4 \Rightarrow T_{3\frac12} \Rightarrow T_3 \Rightarrow T_2 \Rightarrow T_1 \Rightarrow T_0$	each condition implies the one to its right; no arrow reverses	Metrisation and Separation Axioms
compact Hausdorff $\Rightarrow$ normal	every compact Hausdorff space is normal, hence satisfies Urysohn and Tietze	Topological Spaces
metric $\Rightarrow$ perfectly normal, collectionwise normal	every metrisable space is perfectly normal and collectionwise normal	Metrisation and Separation Axioms
paracompact Hausdorff $\Rightarrow$ normal	paracompactness supplies the shrinkings from which the separation is built	Paracompactness and Partitions of Unity
normal $\Rightarrow$ Tietze, Urysohn	the extension and separation theorems hold in every normal space	Metrisation and Separation Axioms

The Spaces That Separate the Axioms

The axioms do not collapse, and each collapse is prevented by a space of the corpus. The failures are recorded beside the spaces that satisfy the axiom, in the same table.

Space	$T_0$	$T_1$	Hausdorff	regular	completely regular	normal	metrisable
$\mathbb{R}$, $\mathbb{R}^n$, $\mathbb{Q}$	yes	yes	yes	yes	yes	yes	yes
$\mathbb{C}$, $\mathbb{C}^n$	yes	yes	yes	yes	yes	yes	yes
$S^n$, $T^n$, $\mathbb{RP}^n$, $\mathbb{CP}^n$	yes	yes	yes	yes	yes	yes	yes
the Cantor set, $\mathbb{Z}_p$, a compact metric space	yes	yes	yes	yes	yes	yes	yes
a compact Hausdorff space	yes	yes	yes	yes	yes	yes	not always
a paracompact Hausdorff space	yes	yes	yes	yes	yes	yes	not always
a metric space	yes	yes	yes	yes	yes	yes	yes
$\beta\mathbb{N}$	yes	yes	yes	yes	yes	yes	no
the long line $L$, $[0,\omega_1)$, $[0,\omega_1]$	yes	yes	yes	yes	yes	yes	no
the Sorgenfrey line $\mathbb{R}_S$	yes	yes	yes	yes	yes	yes	no
the Sorgenfrey plane $\mathbb{R}_S^2$	yes	yes	yes	yes	yes	no	no
the Niemytzki plane	yes	yes	yes	yes	yes	no	no
Mysior's example	yes	yes	yes	yes	no	no	no
the cofinite topology on an infinite set	yes	yes	no	no	no	no	no
the indiscrete topology on two points	no	no	no	yes	yes	yes	no
$[0,1]^{I}$ for uncountable $I$	yes	yes	yes	yes	yes	no	no
$\operatorname{Spec} R$ with the Zariski topology	yes	no	no	—	—	—	no

The rows are read as follows. The cofinite topology is $T_1$ and not Hausdorff, and since $T_1$ together with normality implies Hausdorff, it is neither regular nor normal; this is the counterexample of Topological Spaces. The Sorgenfrey plane and the Niemytzki plane are Tychonoff and not normal, so $T_{3\frac12}$ does not imply $T_4$, and the Sorgenfrey plane is a finite product of normal spaces that is not normal. Mysior's example is regular and not completely regular, so $T_3$ does not imply $T_{3\frac12}$. The indiscrete topology is regular, completely regular and normal and not $T_1$, which is why the corpus defines $T_3$, $T_{3\frac12}$ and $T_4$ with the $T_1$ clause. The product of uncountably many intervals is compact Hausdorff and not normal, so normality is not productive. The prime spectrum is $T_0$ and not $T_1$ whenever a nonzero prime is contained in another, so it is not metrisable, and this failure is one of separation and not of countability.

The Compactness and Countability Conditions

Condition	Its definition	Introduced in
compact	every open cover has a finite subcover; equivalently every ultrafilter converges	Topological Spaces
countably compact	every countable open cover has a finite subcover; equivalently every sequence has a cluster point	Paracompactness and Partitions of Unity
sequentially compact	every sequence has a convergent subsequence; equivalent to compactness in a metric space	Topological Spaces
Lindelöf	every open cover has a countable subcover	Paracompactness and Partitions of Unity
$\sigma$-compact	a countable union of compact subsets	Topological Spaces
locally compact	every point has a neighbourhood base of compact neighbourhoods	Topological Spaces
paracompact	every open cover has a locally finite open refinement	Paracompactness and Partitions of Unity
fully normal	every open cover has an open star refinement; equivalent to paracompactness for Hausdorff spaces	Paracompactness and Partitions of Unity
totally bounded	for every $\varepsilon$ a finite cover by sets of diameter less than $\varepsilon$	Metric, Uniform and Complete Spaces

The implications among the conditions are these, and each is recorded with the article that proves it:

Implication	The statement	Introduced in
compact $\Rightarrow$ countably compact	a finite subcover is a finite subcover of every countable subfamily	Topological Spaces
compact $\Rightarrow$ Lindelöf	a finite subcover is countable	Paracompactness and Partitions of Unity
compact $\Rightarrow$ $\sigma$-compact	the space itself is the single compact set	Topological Spaces
compact Hausdorff $\Rightarrow$ paracompact	a finite subcover is locally finite	Paracompactness and Partitions of Unity
paracompact + countably compact $\Rightarrow$ compact	a locally finite family of nonempty sets is finite in a countably compact space	Paracompactness and Partitions of Unity
countably compact + Lindelöf $\Rightarrow$ compact	the countable subcover has a finite subcover	Paracompactness and Partitions of Unity
metric: compact $\Leftrightarrow$ sequentially compact $\Leftrightarrow$ complete + totally bounded	the metric characterisation of compactness	Topological Spaces; Metric, Uniform and Complete Spaces
metric $\Rightarrow$ paracompact (Stone)	every metrisable space is paracompact	Paracompactness and Partitions of Unity
Lindelöf + regular $\Rightarrow$ paracompact	the countable refinement is locally finite	Paracompactness and Partitions of Unity

The Matrix of the Standard Spaces

The following table records the compactness and countability conditions satisfied by the spaces of the corpus. A blank cell is not a claim of failure; it records that the corpus does not establish the condition for that space.

Space	compact	countably compact	Lindelöf	$\sigma$-compact	locally compact	paracompact
$\mathbb{R}$, $\mathbb{R}^n$, $\mathbb{C}^n$	no	no	yes	yes	yes	yes
$[0,1]$, $S^n$, $T^n$, $\mathbb{RP}^n$	yes	yes	yes	yes	yes	yes
the Cantor set, $\mathbb{Z}_p$	yes	yes	yes	yes	yes	yes
$\mathbb{Q}$	no	no	yes	yes	no	yes
$\mathbb{Z}$, $\mathbb{N}$	no	no	yes	yes	yes	yes
the long line $L$	no	yes	no	no	yes	no
$[0,\omega_1)$	no	yes	no	no	yes	no
$[0,\omega_1]$	yes	yes	yes	yes	yes	yes
$\beta\mathbb{N}$	yes	yes	yes	yes	yes	yes
the Sorgenfrey line $\mathbb{R}_S$	no	no	yes	no	no	yes
the Sorgenfrey plane $\mathbb{R}_S^2$	no	—	no	no	no	no
the Niemytzki plane	no	—	no	no	no	no
a metric space	not always	not always	not always	not always	not always	yes
a compact Hausdorff space	yes	yes	yes	yes	yes	yes

The three standard separations of the table are the long line, the Sorgenfrey line and the Sorgenfrey plane. The long line is countably compact and not compact, and it is locally metrisable and not paracompact: the cover by its initial segments has no locally finite refinement. The Sorgenfrey line is Lindelöf, hence paracompact and normal, and it is not $\sigma$-compact, not locally compact and not metrisable; it shows that Lindelöf and paracompact do not give a distance. The Sorgenfrey plane is the product of two paracompact spaces and is not paracompact, and it is not Lindelöf; it shows that paracompactness is not finitely productive.

Warnings

Objects that a reader may expect to find among the spaces classified here, and does not.

Object	Why it is not tabulated	Introduced in
the Zariski topology alone	a topology that is $T_0$ and not $T_1$; it is recorded through $\operatorname{Spec} R$ and the real Zariski topology, and it carries no metric	Schemes; Real Algebraic Geometry
pseudocompactness	the corpus does not introduce pseudocompact or limit-point compact spaces, and no entry relies on them	—
countable compactness of a general non-Hausdorff space	the corpus records countable compactness only where the paracompactness theory uses it, and not for the pathological spaces	Paracompactness and Partitions of Unity
the axiom $T_5$	the corpus uses the separation hierarchy only up to $T_4$ and perfect normality	Metrisation and Separation Axioms

Summary

This article has listed the separation axioms $T_0$ to $T_4$, the complete regularity and perfect normality that sit beside them, and the compactness and countability conditions of the corpus, and has tabulated which spaces satisfy which. The implications run $T_4 \Rightarrow T_{3\frac12} \Rightarrow T_3 \Rightarrow T_2 \Rightarrow T_1 \Rightarrow T_0$ with no arrow reversing, compactness implies countable compactness, the Lindelöf property and $\sigma$-compactness, and countable compactness together with the Lindelöf property gives compactness. Each non-implication is witnessed by a space of the corpus: the cofinite topology for $T_1$ without Hausdorff, Mysior's example for regularity without complete regularity, the Sorgenfrey plane and the Niemytzki plane for Tychonoff without normal, the indiscrete space for normal without $T_1$, and the long line and the Sorgenfrey line for the independence of compactness, paracompactness and metrisability. The tables introduce and prove nothing; they record the properties that the introducing articles establish.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following.

Symbol	Meaning
$T_0$, $T_1$, $T_2$, $T_3$, $T_{3\frac12}$, $T_4$	the separation axioms; $T_2$ is Hausdorff, $T_3$ regular, $T_{3\frac12}$ Tychonoff, $T_4$ normal, each with the $T_1$ clause
$T_3$, $T_{3\frac12}$	the corpus's numbering, in which complete regularity is stronger than regularity
regular, completely regular, normal, paracompact	the conditions as in Metrisation and Separation Axioms and Paracompactness and Partitions of Unity
$\mathbb{R}_S$, $\mathbb{R}_S^2$	the Sorgenfrey line and its square
$\beta\mathbb{N}$	the Stone–Čech compactification of $\mathbb{N}$
$L$, $[0,\omega_1)$, $[0,\omega_1]$	the long line and the two ordinal spaces
$\operatorname{Spec} R$	the prime spectrum with the Zariski topology
$[0,1]^{I}$	a product of intervals, compact Hausdorff for every $I$
$\sigma$-compact, Lindelöf	countable union of compact sets; every open cover has a countable subcover