Closure of Intersection is Subset of Intersection of Closures

Theorem
Let $T$ be a topological space.

Let $I$ be an indexing set.

Let $\forall i \in I: H_i \subseteq T$.

Then:
 * $\displaystyle \left({\bigcap_I H_i}\right)^- \subseteq \bigcap_I H_i^-$

where $H_i^-$ denotes the closure of $H_i$.

Proof
Since $\displaystyle \bigcap_I H_i^-$ is an intersection of closed sets, it is closed, from Topology Defined by Closed Sets.

Also, it contains $\displaystyle \bigcap_I H_i$ and so by the main definition of closure also contains $\displaystyle \left({\bigcap_I H_i}\right)^-$.

Also see

 * Closure of Intersection may not equal Intersection of Closures, which shows that equality does not generally hold.