# Definition:T3 Space/Definition 2

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

$T = \struct {S, \tau}$ 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 $x$.

## 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.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms