Intersection of Interiors contains Interior of Intersection

Theorem
Let $T$ be a topological space.

Let $\mathbb H$ be a set of subsets of $T$.

That is, let $\mathbb H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$.

Then the interior of the intersection of $\mathbb H$ is a subset of the intersection of the interiors of the elements of $\mathbb H$.
 * $\ds \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ \subseteq \bigcap_{H \mathop \in \mathbb H} H^\circ$

Proof
In the following, $H^-$ denotes the closure of the set $H$.

At this point we note that:
 * $(1): \quad \ds \paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H} }^- \supseteq \bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H}^-$

from Closure of Union contains Union of Closures.

Then we note that:
 * $\ds T \setminus \paren {\paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H} }^-} \subseteq T \setminus \paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H}^-}$

from $(1)$ and Set Complement inverts Subsets.

Then we continue:

Also see

 * Interior of Finite Intersection equals Intersection of Interiors
 * Interior of Intersection may not equal Intersection of Interiors