Exterior of Intersection contains Union of Exteriors

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:
 * $\displaystyle \bigcup_{H \mathop \in \mathbb H} H^e \subseteq \paren {\bigcap_{H \mathop \in \mathbb H} H}^e$

where $H^e$ denotes the exterior of $H$.

Proof
We have:

From Closure of Intersection is Subset of Intersection of Closures:
 * $\displaystyle \paren {\bigcap_{H \mathop \in \mathbb H} H^-} \subseteq \bigcap_{H \mathop \in \mathbb H} H^-$

From Set Complement inverts Subsets:
 * $\displaystyle T \setminus \paren {\bigcap_{H \mathop \in \mathbb H} H^-} \supseteq T \setminus \bigcap_{H \mathop \in \mathbb H} H^-$

From the definition of exterior:
 * $\displaystyle T \setminus \paren {\bigcup_{H \mathop \in \mathbb H} H}^- = \paren {\bigcup_{H \mathop \in \mathbb H} H}^e$

Putting this together:
 * $\displaystyle \bigcup_{H \mathop \in \mathbb H} H^e \subseteq \paren {\bigcap_{H \mathop \in \mathbb H} H}^e$