Exterior of Finite Union equals Intersection of Exteriors

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