Union of Interiors is Subset of Interior of Union

Theorem
Let $T$ be a topological space.

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

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

Then the union of the interiors of the elements of $\H$ is a subset of the interior of the union of $\H$:
 * $\ds \bigcup_{H \mathop \in \H} H^\circ \subseteq \paren {\bigcup_{H \mathop \in \H} H}^\circ $

Also see

 * Interior of Union is not necessarily Union of Interiors