Definition:T5 Space/Definition 2

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

$\struct {S, \tau}$ is a $T_5$ space if and only if:

$\forall Y,A \subseteq S: (A \subseteq Y^\circ \wedge A^- \subseteq Y) \implies \exists N \subseteq Y: \relcomp S N \in \tau: \exists U \in \tau: A \subseteq U \subseteq N$

That is:

$\struct {S, \tau}$ is a $T_5$ space if and only if every subset $Y \subseteq S$ contains a closed neighborhood of each $A \subseteq Y^\circ$ for which $A^- \subseteq Y$.

In the above, $Y^\circ$ denotes the interior of $Y$ and $A^-$ denotes the closure of $A$.

Variants of Name

From about 1970, treatments of this subject started to refer to this as a completely normal space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a completely normal space as a $T_5$ 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 $T_5$ spaces can be found here.