Open Set Disjoint from Set is Disjoint from Closure

Theorem
Let $\left({T, \tau}\right)$ be a topological space.

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

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

where $\operatorname{cl} \left({A}\right)$ denotes the closure of $A$.

Proof
Since $B \in \tau$, it follows by definition that $T \setminus B$ is closed.

Thus: