Topological Exterior#

Definition: Exterior Point

Let \(S \subseteq X\) be a subset of a topological space \(X\).

We say that \(p \in X\) is an exterior point of \(S\) if it has a neighborhood which is disjoint from \(S\).

\[\exists N(p): N(p) \cap S = \varnothing\]

Definition: Topological Exterior

The (topological) exterior of \(S\) is the set of all its exterior points.

Notation

\[\operatorname{ext} S \qquad \operatorname{ext}_X S\]

Theorem: Exterior via Open Sets

Let \(S \subseteq X\) be a subset of a topological space \(X\).

The exterior of \(S\) is the union of all open sets which are disjoint from \(S\):

\[\operatorname{ext} S = \bigcup \{U \subseteq X \mid U \text{ is open and } U \cap S = \varnothing\}\]

Proof

TODO