Local Basis Generated from Neighborhood Basis

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

Let $x$ be an element of $S$.

Let $\mathcal B$ be a neighborhood basis of $x$.

For any subset $A \subseteq S$, let $A^\circ$ denote the interior of $A$.

Then the set:
 * $\mathcal {B’} = \set{H^\circ : H \in B}$

is a local basis of $x$.