# Definition:Completely Normal Space

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## Definition

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

$\struct {S, \tau}$ is a **completely normal space** if and only if:

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

That is:

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

- $\forall x, y \in S$,
*both*:- $\exists U \in \tau: x \in U, y \notin U$
- $\exists V \in \tau: y \in V, x \notin V$

## Variants of Name

From about 1970, treatments of this subject started to refer to this as a **$T_5$ space**, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a $T_5$ space as a **completely normal 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
**completely normal 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: Regular and Normal Spaces - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**T-axioms**or**Tychonoff conditions**:**5.**(**$T_5$ space**)