Interior of Finite Intersection equals Intersection of Interiors

Theorem

Let $T$ be a topological space.

Let $n \in \N$.

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

Then:

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

where $H_i^\circ$ denotes the interior of $H_i$.

Proof

In the following, $H_i^-$ denotes the closure of the set $H_i$.

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

$\blacksquare$

Sources

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