Intersections of open sets

Question

Suppose $X$ is a topological space such that $$\bigcap_{i\in I} \overline{U_i} = \overline{ \textrm{int}\bigcap_{i\in I} U_i }$$ holds for every family of open sets $\{U_i\}_{i\in I}$ in $X$. I would like to know if this class of spaces has a name or any other (more useful) characterization.

Answer 1

As Kavi pointed out, a necessary condition is
that the space be extremally disconnected.

An Alexander space is a space for which
any intersection of open sets is open.
A sufficient condition is that the space
be an Alexander space.