Union of Interiors is Subset of Interior of Union/Proof 1

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

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

from Closure of Intersection is Subset of Intersection of Closures.

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

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

Then we continue: