Definition:T5 Space/Definition 2

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

$\struct {S, \tau}$ is a $T_5$ space :
 * $\forall Y, A \subseteq S: (A \subseteq Y^\circ \wedge A^- \subseteq Y) \implies \exists N \subseteq Y: \relcomp S N \in \tau: \exists U \in \tau: A \subseteq U \subseteq N$

That is:
 * $\struct {S, \tau}$ is a $T_5$ space 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 $A$.

Also see

 * Equivalence of Definitions of $T_5$ Space