Closure of Intersection Given a topological space $\Omega$.
By minimality it holds:
$$\overline{A\cap B}\subseteq\overline{A}\cap\overline{B}$$
Especially one has equality for:
$$A=\overline{A}:\quad\overline{A\cap B}=A\cap\overline{B}$$
How to check this quickly?

What if $A= [0,1]$ and $B=(1,2),$ then $A \cap B = \emptyset$ so $\overline{A \cap B} = \emptyset,$ but $A \cap \overline{B}= [0,1] \cap [1,2]=\{1\}.$ So you don't have the equality.
