Interior of Finite Intersection equals Intersection of Interiors
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Theorem
Let $T$ be a topological space.
Let $n \in \N$.
Let:
- $\forall i \in \set {1, 2, \dotsc, n}: H_i \subseteq T$
Then:
- $\ds \paren {\bigcap_{i \mathop = 1}^n H_i}^\circ = \bigcap_{i \mathop = 1}^n H_i^\circ$
where $H_i^\circ$ denotes the interior of $H_i$.
Proof
In the following, $H_i^-$ denotes the closure of the set $H_i$.
\(\ds \paren {\bigcap_{i \mathop = 1}^n H_i}^\circ\) | \(=\) | \(\ds T \setminus \paren {T \setminus \bigcap_{i \mathop = 1}^n H_i}^-\) | Complement of Interior equals Closure of Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds T \setminus \paren {\paren {\bigcup_{i \mathop = 1}^n \paren {T \setminus H_i} }^-}\) | De Morgan's Laws: Difference with Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds T \setminus \paren {\bigcup_{i \mathop = 1}^n \paren {T \setminus H_i}^-}\) | Closure of Finite Union equals Union of Closures | |||||||||||
\(\ds \) | \(=\) | \(\ds T \setminus \paren {\bigcup_{i \mathop = 1}^n T \setminus H_i^\circ}\) | Complement of Interior equals Closure of Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds T \setminus \paren {T \setminus \paren {\bigcap_{i \mathop = 1}^n H_i^\circ} }\) | De Morgan's Laws: Difference with Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{i \mathop = 1}^n H_i^\circ\) | Relative Complement of Relative Complement |
$\blacksquare$
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