Definition:Perfectly Normal Space
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a perfectly normal space if and only if:
- $\struct {S, \tau}$ is a perfectly $T_4$ space
- $\struct {S, \tau}$ is a $T_1$ (Fréchet) space.
That is:
- Every closed set in $T$ is a $G_\delta$ set.
- $\forall x, y \in S$, both:
- $\exists U \in \tau: x \in U, y \notin U$
- $\exists V \in \tau: y \in V, x \notin V$
Also see
- Results about perfectly 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: Additional Separation Properties