Intersection of Exteriors contains Exterior of Union

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Theorem

Let $T$ be a topological space.


Let $\mathbb H$ be a set of subsets of $T$.

That is, let $\mathbb H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$.


Then:

$\ds \paren {\bigcup_{H \mathop \in \mathbb H} H}^e \subseteq \bigcap_{H \mathop \in \mathbb H} H^e $

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


Proof

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

\(\ds \paren {\bigcup_{H \mathop \in \mathbb H} H}^e\) \(=\) \(\ds \paren {T \setminus \bigcup_{H \mathop \in \mathbb H} H}^\circ\) Definition of Exterior
\(\ds \) \(=\) \(\ds \paren {\bigcap_{H \mathop \in \mathbb H} \paren {T \setminus H} }^\circ\) De Morgan's Laws: Difference with Union
\(\ds \) \(\subseteq\) \(\ds \bigcap_{H \mathop \in \mathbb H} \paren {T \setminus H}^\circ\) Intersection of Interiors contains Interior of Intersection
\(\ds \) \(=\) \(\ds \bigcap_{H \mathop \in \mathbb H} H^e\) Definition of Exterior

$\blacksquare$


Sources