Equivalence of Definitions of T3 Space

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

The following three conditions defining a $T_3$ space are logically equivalent:

Definition by Open Sets implies Definition by Closed Neighborhoods
Let $T = \left({S, \tau}\right)$ be a topological space for which:


 * $\forall F \subseteq S: \complement_S \left({F}\right) \in \tau, y \in \complement_S \left({F}\right): \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \varnothing$

Definition by Closed Neighborhoods implies Definition by Open Sets
$T = \left({S, \tau}\right)$ is a topological space for which:


 * $\forall U \in \tau: \forall x \in U: \exists N_x: \complement_S \left({N_x}\right) \in \tau: \exists V \in \tau: x \in V \subseteq N_x \subseteq U$