Definition:Normal Space

Definition

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

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

$\struct {S, \tau}$ is a $T_4$ space

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

That is:

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

$\forall x, y \in S$, 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 $T_4$ space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a $T_4$ space as a 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.

This space is also referred to as normal Hausdorff.

Also see

• Results about 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): T-axioms or Tychonoff conditions: 4. ($T_4$ space)