Intersection of Interiors contains Interior of Intersection
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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 the interior of the intersection of $\mathbb H$ is a subset of the intersection of the interiors of the elements of $\mathbb H$.
- $\ds \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ \subseteq \bigcap_{H \mathop \in \mathbb H} H^\circ$
Proof
In the following, $H^-$ denotes the closure of the set $H$.
\(\ds \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ\) | \(=\) | \(\ds T \setminus \paren {T \setminus \bigcap_{H \mathop \in \mathbb H} H}^-\) | Complement of Interior equals Closure of Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds T \setminus \paren {\paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H} }^-}\) | De Morgan's Laws: Difference with Intersection |
At this point we note that:
- $(1): \quad \ds \paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H} }^- \supseteq \bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H}^-$
from Closure of Union contains Union of Closures.
Then we note that:
- $\ds T \setminus \paren {\paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H} }^-} \subseteq T \setminus \paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H}^-}$
from $(1)$ and Set Complement inverts Subsets.
Then we continue:
\(\ds T \setminus \paren {\bigcup_{H \mathop \in \mathbb H} \paren {T \setminus H}^-}\) | \(=\) | \(\ds T \setminus \paren {\bigcup_{H \mathop \in \mathbb H} T \setminus H^\circ}\) | Complement of Interior equals Closure of Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds T \setminus \paren {T \setminus \paren {\bigcap_{H \mathop \in \mathbb H} H^\circ} }\) | De Morgan's Laws: Difference with Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{H \mathop \in \mathbb H} H^\circ\) | Relative Complement of Relative Complement |
$\blacksquare$
Also see
- Interior of Finite Intersection equals Intersection of Interiors
- Interior of Intersection may not equal Intersection of Interiors
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