User:Leigh.Samphier/Topology/Characterization of T1 Space using Neighborhood Basis

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

For each $x \in S$, let $\NN_x$ be a neighborhood basis at $x$.

Then:
 * $T$ is a $T_1$ Space


 * $\forall x, y \in S : x \ne y$, both:
 * $\exists N \in \NN_x : y \notin N$
 * and:
 * $\exists M \in \NN_y : x \notin M$