Finite Intersection of Regular Open Sets is Regular Open

Theorem

Let $T$ be a topological space.

Let $n \in \N$.

Suppose that:

$\forall i \in \left[{1 .. n}\right]: H_i \subseteq T$

where all the $H_i$ are regular open in $T$, i.e.:

$\forall i \in \left[{1 .. n}\right]: H_i = H_i^{- \circ}$

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

That is:

$\displaystyle \bigcap_{i=1}^n H_i = \left({\bigcap_{i=1}^n H_i}\right)^{- \circ}$

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({\bigcap_{i=1}^n H_i}\right)^{- \circ}\)	\(=\)	\(\displaystyle \left({T \setminus \left({T \setminus \bigcap_{i=1}^n H_i}\right)^\circ}\right)^\circ\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Complement of Interior equals Closure of Complement
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({T \setminus \left({\bigcup_{i=1}^n \left({T \setminus H_i}\right)}\right)^\circ}\right)^\circ\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	De Morgan's Laws
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\left({T \setminus \bigcup_{i=1}^n \left({T \setminus H_i}\right)}\right)^-}\right)^\circ\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Complement of Interior equals Closure of Complement
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({ \bigcap_{i=1}^n \left({T \setminus \left({T \setminus H_i}\right)}\right)^-}\right)^\circ\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	De Morgan's Laws
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\bigcap_{i=1}^n H_i^-}\right)^\circ\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Relative Complement of Relative Complement
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \bigcap_{i=1}^n H_i^{- \circ}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Interior of Finite Intersection equals Intersection of Interiors
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \bigcap_{i=1}^n H_i\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	as all $H_i$ are regular open in $T$

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Closures and Interiors