# Definition:T4 Space

## Definition

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

### Definition 1

$T = \struct {S, \tau}$ is a **$T_4$ space** if and only if:

- $\forall A, B \in \map \complement \tau, A \cap B = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$

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.

### Definition 2

$T = \struct {S, \tau}$ is **$T_4$** if and only if each open set $U$ contains a closed neighborhood of each closed set contained in $U$.

## Variants of Name

From about 1970, treatments of this subject started to refer to this as a **normal space**, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a normal space as a **$T_4$ 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
**$T_4$ spaces**can be found**here**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**normal topological space**