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
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors