Definition:T3 Space/Definition 1

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

$T = \struct {S, \tau}$ is a $T_3$ space :


 * $\forall F \subseteq S: \relcomp S F \in \tau, y \in \relcomp S F: \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \O$

That is, for any closed set $F \subseteq S$ and any point $y \in S$ such that $y \notin F$ there exist disjoint open sets $U, V \in \tau$ such that $F \subseteq U$, $y \in V$.

That is:
 * $\struct {S, \tau}$ is $T_3$ when any closed set $F \subseteq S$ and any point not in $F$ are separated by neighborhoods.

Also see

 * Equivalence of Definitions of $T_3$ Space