Exterior of Finite Union equals Intersection of Exteriors
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Theorem
Let $T$ be a topological space.
Let $n \in \N$.
Let $\forall i \in \closedint 1 n: H_i \subseteq T$.
Then:
- $\ds \paren {\bigcup_{i \mathop = 1}^n H_i}^e = \bigcap_{i \mathop = 1}^n H_i^e$
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 |
$\blacksquare$
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