Open Set Disjoint from Set is Disjoint from Closure

Theorem
Let $(T,\tau)$ be a topological space.

Let $A, B \in \tau$ such that $A \cap B = \varnothing$.

Then:
 * $\operatorname{cl}(A) \cap B = \varnothing$

Proof
Since $B \in \tau$, $\ T\setminus B$ is closed.

Since $A \cap B = \varnothing$, $A \subseteq T \setminus B$ and $\operatorname{cl}(A) \subseteq T \setminus B$.

Now, we easily deduce that $\operatorname{cl}(A)\cap B = \varnothing$.