Definition:T5 Space
Definition
Let $T = \left({S, \tau}\right)$ be a topological space.
Definition 1
$\struct {S, \tau}$ is a $T_5$ space if and only if:
- $\forall A, B \subseteq S, A^- \cap B = A \cap B^- = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$
That is:
- $\struct {S, \tau}$ is a $T_5$ space when for any two separated sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau$ containing $A$ and $B$ respectively.
Definition 2
$\struct {S, \tau}$ 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.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): normal topological space