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

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

Let $\BB$ be a basis for $T$.

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


 * $\forall x, y \in S : x \ne y$, both:
 * $\exists B_x \in \BB : x \in B_x, y \notin B_x$
 * and:
 * $\exists B_y \in \BB : y \in B_y, x \notin B_y$