$\overline{U}\cap V\subseteq\overline{U\cap V}$ In particular if the equality is generally false, is it true if $V$ is open?
Could someone help, me please? 

In general this is false: 
Consider $U := [0,1)$ and $V:= [1,2]$. The we have $\overline{U} = [0,1]$ and thus $\overline{U} \cap V = \{1\}$ but $U \cap V = \emptyset$ and thus their closure is empty as well.
However, if $V$ is open, this is true.
