Definition:T4 Space/Definition 1

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

$T = \struct {S, \tau}$ is a $T_4$ space :


 * $\forall A, B \in \map \complement \tau, A \cap B = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$

That is, for any two disjoint closed sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau$ containing $A$ and $B$ respectively.

That is:
 * $T = \struct {S, \tau}$ is $T_4$ when any two disjoint closed subsets of $S$ are separated by neighborhoods.

Also see

 * Equivalence of Definitions of $T_4$ Space