Definition:T5 Space

Definition
Let $T = \left({X, \vartheta}\right)$ be a topological space.

$\left({X, \vartheta}\right)$ is a $T_5$ space iff:


 * $\forall A, B \subseteq X, A^- \cap B = A \cap B^- = \varnothing: \exists U, V \in \vartheta: A \subseteq U, B \subseteq V, U \cap V = \varnothing$

That is:
 * $\left({X, \vartheta}\right)$ is a $T_5$ space when for any two separated sets $A, B \subseteq X$ there exist disjoint open sets $U, V \in \vartheta$ containing $A$ and $B$ respectively.

Equivalent Definitions

 * $\left({X, \vartheta}\right)$ is a $T_5$ space iff each subset $Y$ contains a closed neighborhood of each $A \subseteq Y^\circ$ where $A^- \subseteq Y$.

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

This is proved in Equivalent Definitions for $T_5$ Space.