Characterization of Boundary by Basis

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

Let $\mathcal B \subseteq \tau$ be a basis.

Let $A$ be a subset of $T$.

Let $x$ be a point of $T$.

Then $x \in \operatorname{fr} A$ iff for every subset $U \in \mathcal B$ if $x \in U$, then $A \cap U \neq \varnothing$ and $A` \cap U \neq \varnothing$

where $A` = S \setminus A$ stands for complement of $A$.

Proof
To prove first implication follows from Characterization of Boundary by Open Sets because element of basis is open.