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 $\mathbb H \subseteq \mathcal P \left({T}\right)$ where $\mathcal P \left({T}\right)$ is the power set of $T$.

Then:
 * $\displaystyle \bigcup_{H \in \mathbb H} H^e \subseteq \left({\bigcap_{H \in \mathbb H} H}\right)^e$

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

Proof
We have:

From Closure of Intersection Subset of Intersection of Closures:
 * $\displaystyle \left({\bigcap_{H \in \mathbb H} H^-}\right) \subseteq \bigcap_{H \in \mathbb H} H^-$

From Complements Invert Subsets:
 * $\displaystyle T \setminus \left({\bigcap_{H \in \mathbb H} H^-}\right) \supseteq T \setminus \bigcap_{H \in \mathbb H} H^-$

From the definition of exterior:
 * $\displaystyle T \setminus \left({\bigcup_{H \in \mathbb H} H}\right)^- = \left({\bigcup_{H \in \mathbb H} H}\right)^e$

Putting this together:
 * $\displaystyle \bigcup_{H \in \mathbb H} H^e \subseteq \left({\bigcap_{H \in \mathbb H} H}\right)^e$

Mistakes in Sources
See : $\text{I}: \S 1$ where it is stated that:
 * Union of Exteriors contains Exterior of Intersection