Definition:Tychonoff Separation Axioms

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Definition

The Tychonoff separation axioms 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, $T = \left({S, \tau}\right)$ is a topological space with topology $\tau$.


$T_0$ (Kolmogorov) Space

$\left({S, \tau}\right)$ is a Kolmogorov space or $T_0$ space if and only if:

$\forall x, y \in S$ such that $x \ne y$, either:
$\exists U \in \tau: x \in U, y \notin U$
or:
$\exists U \in \tau: y \in U, x \notin U$


$T_1$ (Fréchet) Space

$\left({S, \tau}\right)$ is a Fréchet space or $T_1$ space if and only if:

$\forall x, y \in S$ such that $x \ne y$, both:
$\exists U \in \tau: x \in U, y \notin U$
and:
$\exists V \in \tau: y \in V, x \notin V$


$T_2$ (Hausdorff) Space

$\left({S, \tau}\right)$ is a Hausdorff space or $T_2$ space if and only if:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$

That is:

for any two distinct elements $x, y \in S$ there exist disjoint open sets $U, V \in \tau$ containing $x$ and $y$ respectively.


Semiregular Space

$\left({S, \tau}\right)$ is a semiregular space if and only if:

$\left({S, \tau}\right)$ 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

$\left({S, \tau}\right)$ is a completely Hausdorff space or $T_{2 \frac 1 2}$ space if and only if:

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

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

That is:

$\left({S, \tau}\right)$ is a $T_{2 \frac 1 2}$ space if and only if every two points in $S$ are separated by closed neighborhoods.


$T_3$ Space

$T = \left({S, \tau}\right)$ is a $T_3$ space if and only if:

$\forall F \subseteq S: \complement_S \left({F}\right) \in \tau, y \in \complement_S \left({F}\right): \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \varnothing$

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


Regular Space

$\left({S, \tau}\right)$ is a regular space if and only if:

$\left({S, \tau}\right)$ is a $T_3$ space
$\left({S, \tau}\right)$ is a $T_0$ (Kolmogorov) space.


Urysohn Space

$\left({S, \tau}\right)$ is an Urysohn space if and only if:

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


$T_{3 \frac 1 2}$ Space

$\left({S, \tau}\right)$ is a $T_{3 \frac 1 2}$ space if and only if:

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


Tychonoff (Completely Regular) Space

$\left({S, \tau}\right)$ is a Tychonoff Space or completely regular space if and only if:

$\left({S, \tau}\right)$ is a $T_{3 \frac 1 2}$ space
$\left({S, \tau}\right)$ is a $T_0$ (Kolmogorov) space.


$T_4$ Space

$T = \left({S, \tau}\right)$ is a $T_4$ space if and only if:

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

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


Normal Space

$\left({S, \tau}\right)$ is a normal space if and only if:

$\left({S, \tau}\right)$ is a $T_4$ space
$\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.


$T_5$ Space

$\left({S, \tau}\right)$ is a $T_5$ space if and only if:

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

That is:

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


Completely Normal Space

$\left({S, \tau}\right)$ is a completely normal space if and only if:

$\left({S, \tau}\right)$ is a $T_5$ space
$\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.


Perfectly $T_4$ Space

$T$ is a perfectly $T_4$ space if and only if:

$(1): \quad T$ 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

$\left({S, \tau}\right)$ is a perfectly normal space if and only if:

$\left({S, \tau}\right)$ is a perfectly $T_4$ space
$\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.


Naming Conventions

There are different ways of naming the separation axioms. The technique for this site is to follow the convention used in 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology. 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 known as

The Tychonoff separation axioms are also known as the Tychonoff conditions.

Some sources refer to them as just the separation axioms.

Some sources call them the $T_i$ axioms.


Also see

  • Results about the separation axioms can be found here.


Source of Name

This entry was named for Andrey Nikolayevich Tychonoff.


Linguistic Note

The letter $T$ used to denote the Tychonoff separation axioms comes from the German Trennungsaxiom, which means separation axiom.


Sources