general topology, intersection of open sets

Question

under what conditions on a topology we have: $cl(U_1 \cap U_2)=cl(U_1) \cap cl(U_2)$ if $U_1,U_2$ is open

Answer 1

This condition implies that disjoint open sets have disjoint closures, which in turn implies that the closure of an open set is open. This property is called "$X$ is extremally disconnected". It is well known that a metric space that is extremally disconnected is discrete (i.e. all subsets are open). I'm not sure whether any extremally disconnected space will obey the asked for identity, but it's pretty closely related condition anyway. So not too many spaces will satisfy it, though there are classical non-metrisable examples.

Answer 2

Are you looking for any sufficient condition?

The $\subset$ inclusion is the one which could fail ($U_1=(0,1), U_2=(1,2)$), so if you ask the topology, for example, that every open would be closed, then your equality holds for every open sets $U_1, U_2$.