Finite Union of Regular Closed Sets is Regular Closed

Theorem

Let $T$ be a topological space.

Let $n \in \N$.

Suppose that:

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

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

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

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

That is:

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

Proof

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