Finite Intersection of Regular Open Sets is Regular Open

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 open in $T$.

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

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

Then $\displaystyle \bigcap_{i \mathop = 1}^n H_i$ is regular open in $T$.

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