# Definition:T3 Space

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

### Definition 1

$T = \struct {S, \tau}$ is a **$T_3$ space** if and only if:

- $\forall F \subseteq S: \relcomp S F \in \tau, y \in \relcomp S F: \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \O$

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 $F \subseteq U$, $y \in V$.

### Definition 2

$T = \struct {S, \tau}$ is **$T_3$** if and only if each open set contains a closed neighborhood around each of its points:

- $\forall U \in \tau: \forall x \in U: \exists N_x: \relcomp S {N_x} \in \tau: \exists V \in \tau: x \in V \subseteq N_x \subseteq U$

where $N_x$ denotes a neighborhood of $x$.

### Definition 3

$T = \struct {S, \tau}$ is **$T_3$** if and only if each of its closed sets is the intersection of its closed neighborhoods:

- $\forall H \subseteq S: \relcomp S H \in \tau: H = \bigcap \set {N_H: \relcomp S H \in \tau, \exists V \in \tau: H \subseteq V \subseteq N_H}$

## Variants of Name

From about 1970, treatments of this subject started to refer to this as a **regular space**, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a regular space as a **$T_3$ space**.

However, the names are to a fair extent arbitrary and a matter of taste, as there appears to be no completely satisfactory system for naming all these various Tychonoff separation axioms.

The system as used here broadly follows 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: *Counterexamples in Topology* (2nd ed.).

The system used on the Separation axiom page at Wikipedia differs from this.

## Also see

- Results about
**$T_3$ spaces**can be found here.