Finite Union of Regular Closed Sets is Regular Closed

Theorem
Let $T$ be a topological space.

Let $n \in \N$.

Suppose that:
 * $\forall i \in \set {1, 2, \dotsc, n}: H_i \subseteq T$

where all the $H_i$ are regular closed in $T$.

That is:
 * $\forall i \in \set {1, 2, \dotsc, n}: H_i = H_i^{\circ -}$

where $H_i^{\circ -}$ denotes the closure of the interior of $H_i$

Then $\ds \bigcup_{i \mathop = 1}^n H_i$ is regular closed in $T$.

That is:
 * $\ds \bigcup_{i \mathop = 1}^n H_i = \paren {\bigcup_{i \mathop = 1}^n H_i}^{\circ -}$