Definition:Separation Axioms

Definition

The Separation Axioms (sometimes known as the Tychonoff Separation Axioms, for Andrey Nikolayevich Tychonoff) are a classification system for topological spaces.

They are not axiomatic as such, but conditions that may or may not apply to general or specific topological spaces.

In general, each condition is stronger than the previous one, with subtleties.

For all of these definitions, $T = \left({X, \vartheta}\right)$ is a topological space with topology $\vartheta$.

$T_0$ (Kolmogorov) Space

$\left({X, \vartheta}\right)$ is a Kolmogorov space or $T_0$ space iff:

$\forall x, y \in X$ such that $x \ne y$, either:

$\exists U \in \vartheta: x \in U, y \notin U$

or:

$\exists U \in \vartheta: y \in U, x \notin U$

That is, for any two distinct points $x, y \in X$ there exists an open set $U \in \vartheta$ which contains one of the points, but not the other.

That is:

$\left({X, \vartheta}\right)$ is a $T_0$ space iff every two points in $X$ are topologically distinguishable.

$T_1$ (Fréchet) Space

$\left({X, \vartheta}\right)$ is a Fréchet space or $T_1$ space iff:

$\forall x, y \in X$ such that $x \ne y$, both:

$\exists U \in \vartheta: x \in U, y \notin U$

and:

$\exists V \in \vartheta: y \in V, x \notin V$

That is, for any two distinct points $x, y \in X$ there exist open sets $U, V \in \vartheta$ such that $x$ is in $U$ but not in $V$, and $y$ is in $V$ but not in $U$.

That is:

$\left({X, \vartheta}\right)$ is $T_1$ when every two points in $X$ are separated.

$T_2$ (Hausdorff) Space

$\left({X, \vartheta}\right)$ is a Hausdorff space or $T_2$ space iff:

$\forall x, y \in X, x \ne y: \exists U, V \in \vartheta: x \in U, y \in V: U \cap V = \varnothing$

That is, for any two distinct points $x, y \in X$ there exist disjoint open sets $U, V \in \vartheta$ containing $x$ and $y$ respectively.

That is:

$\left({X, \vartheta}\right)$ is a $T_2$ space iff every two points in $X$ are separated by neighborhoods.

Semiregular Space

$\left({X, \vartheta}\right)$ is a semiregular space iff:

$\left({X, \vartheta}\right)$ is a Hausdorff ($T_2$) space

The regular open sets of $T$ form a basis for $T$.

$T_{2 \frac 1 2}$ (Completely Hausdorff) Space

$\left({X, \vartheta}\right)$ is a completely Hausdorff space or $T_{2 \frac 1 2}$ space iff:

$\forall x, y \in X, x \ne y: \exists U, V \in \vartheta: x \in U, y \in V: U^- \cap V^- = \varnothing$

That is, for any two distinct points $x, y \in X$ there exist open sets $U, V \in \vartheta$ containing $x$ and $y$ respectively whose closures are disjoint.

That is:

$\left({X, \vartheta}\right)$ is a $T_{2 \frac 1 2}$ space iff every two points in $X$ are separated by closed neighborhoods.

$T_3$ Space

$T = \left({X, \vartheta}\right)$ is a $T_3$ space iff:

$\forall F \subseteq X: \complement_X \left({F}\right) \in \vartheta, y \in \complement_X \left({F}\right): \exists U, V \in \vartheta: F \subseteq U, y \in V: U \cap V = \varnothing$

That is, for any closed set $F \subseteq X$ and any point $y \in X$ such that $y \notin F$ there exist disjoint open sets $U, V \in \vartheta$ such that $F \subseteq U$, $y \in V$.

That is:

$\left({X, \vartheta}\right)$ is $T_3$ when any closed set $F \subseteq X$ and any point not in $F$ are separated by neighborhoods.

Regular Space

$\left({X, \vartheta}\right)$ is a regular space iff:

$\left({X, \vartheta}\right)$ is a $T_3$ space

$\left({X, \vartheta}\right)$ is a $T_0$ (Kolmogorov) space.

Urysohn Space

$\left({X, \vartheta}\right)$ is an Urysohn space iff:

For any distinct points $x, y \in X$ (i.e. $x \ne y$), there exists an Urysohn function for $\left\{{x}\right\}$ and $\left\{{y}\right\}$.

$T_{3 \frac 1 2}$ Space

$\left({X, \vartheta}\right)$ is a $T_{3 \frac 1 2}$ space iff:

For any closed set $F \subseteq X$ and any point $y \in X$ such that $y \notin F$, there exists an Urysohn function for $F$ and $\left\{{y}\right\}$.

Tychonoff (Completely Regular) Space

$\left({X, \vartheta}\right)$ is a Tychonoff Space or completely regular space iff:

$\left({X, \vartheta}\right)$ is a $T_{3 \frac 1 2}$ space

$\left({X, \vartheta}\right)$ is a $T_0$ (Kolmogorov) space.

$T_4$ Space

$T = \left({X, \vartheta}\right)$ is a $T_4$ space iff:

$\forall A, B \in \complement \left({\vartheta}\right), A \cap B = \varnothing: \exists U, V \in \vartheta: A \subseteq U, B \subseteq V, U \cap V = \varnothing$

That is, for any two disjoint closed sets $A, B \subseteq X$ there exist disjoint open sets $U, V \in \vartheta$ containing $A$ and $B$ respectively.

That is:

$T = \left({X, \vartheta}\right)$ is $T_4$ when any two disjoint closed subsets of $X$ are separated by neighborhoods.

Normal Space

$\left({X, \vartheta}\right)$ is a normal space iff:

$\left({X, \vartheta}\right)$ is a $T_4$ space

$\left({X, \vartheta}\right)$ is a $T_1$ (Fréchet) space.

$T_5$ Space

$\left({X, \vartheta}\right)$ is a $T_5$ space iff:

$\forall A, B \subseteq X, A^- \cap B = A \cap B^- = \varnothing: \exists U, V \in \vartheta: A \subseteq U, B \subseteq V, U \cap V = \varnothing$

That is:

$\left({X, \vartheta}\right)$ is a $T_5$ space when for any two separated sets $A, B \subseteq X$ there exist disjoint open sets $U, V \in \vartheta$ containing $A$ and $B$ respectively.

Completely Normal Space

$\left({X, \vartheta}\right)$ is a completely normal space iff:

$\left({X, \vartheta}\right)$ is a $T_5$ space

$\left({X, \vartheta}\right)$ is a $T_1$ (Fréchet) space.

Perfectly $T_4$ Space

$T$ is a perfectly $T_4$ space iff:

$(1): \quad T$ is a $T_4$ space

$(2): \quad$ Every closed set in $T$ is a $G_\delta$ set.

That is:

Every closed set in $T$ can be written as a countable intersection of open sets of $T$.

Perfectly Normal Space

$\left({X, \vartheta}\right)$ is a perfectly normal space iff:

$\left({X, \vartheta}\right)$ is a perfectly $T_4$ space

$\left({X, \vartheta}\right)$ is a $T_1$ (Fréchet) space.

Linguistic Note

The letter $T$ comes from the German Trennungsaxiom, which means separation axiom.

Naming Conventions

There are different ways of naming the separation axioms. The technique for this site is to follow the convention used in Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970). Beware: this differs from the Separation axiom page at Wikipedia. The various naming schemes are inconsistent with each other and confusing, and no completely satisfactory convention has been defined. It is suggested that the system used here is more modern than others, but there is little evidence one way or another.

An attempt has been made on the appropriate pages to mention the alternative names of these spaces, but this is inconsistent and possibly inaccurate. The important things to note are the conditions themselves and the relations between them. This is a new area of mathematics in which research is ongoing, and the whole area of ground may shift again completely in the near future.

Also see

• Sequence of Implications of Separation Axioms

• Results about the separation axioms can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 2$