Definition:Completely Normal Space

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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.