# Definition:T3 Space/Definition 1

## Definition

Let $T = \left({S, \tau}\right)$ be a topological 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 $F \subseteq U$, $y \in V$.

That is:

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

## 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 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: *Counterexamples in Topology*.

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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 2$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $4.2$: Separation axioms: Definitions $4.2.5$