# Exterior of Finite Union equals Intersection of Exteriors

## Theorem

Let $T$ be a topological space.

Let $n \in \N$.

Let $\forall i \in \closedint 1 n: H_i \subseteq T$.

Then:

$\displaystyle \paren {\bigcup_{i \mathop = 1}^n H_i}^e = \bigcap_{i \mathop = 1}^n H_i^e$

where $H_i^e$ denotes the exterior of $H_i$.

## Proof

In the following, $H_i^\circ$ denotes the interior of the set $H_i$.

 $\ds \paren {\bigcup_{i \mathop = 1}^n H_i}^e$ $=$ $\ds \paren {T \setminus \bigcup_{i \mathop = 1}^n H_i}^\circ$ Definition of Exterior $\ds$ $=$ $\ds \paren {\bigcap_{i \mathop = 1}^n \paren {T \setminus H_i} }^\circ$ De Morgan's Laws: Difference with Union $\ds$ $=$ $\ds \bigcap_{i \mathop = 1}^n \paren {T \setminus H_i}^\circ$ Interior of Finite Intersection equals Intersection of Interiors $\ds$ $=$ $\ds \bigcap_{i \mathop = 1}^n H_i^e$ Definition of Exterior

$\blacksquare$