Topological Interior#
Definition: Interior Point
Let \(S \subseteq X\) be a subset of a topological space \(X\).
We say that \(p \in X\) is an interior point of \(S\) if it has a neighborhood contained in \(S\):
\[\exists N(p): N(p) \subseteq S\]
Definition Topological Interior
The (topological) interior of \(S\) is the set of all its interior points.
Notation
\[\operatorname{int} S \qquad \operatorname{int}_X S \qquad \mathring{S}\]
Theorem: Interior via Open Sets
Let \(S \subseteq X\) be a subset of a topological space \(X\).
The interior of \(S\) is the union of all open sets contained in \(S\):
\[\operatorname{int} S = \bigcup \{U \subseteq X \mid U \text{ is open and } U \subseteq S\}\]
Proof
TODO
Theorem: Interior is a Subset
Let \(S \subseteq X\) be a subset of a topological space \(X\).
The interior of \(S\) is a subset of \(S\):
\[\operatorname{int} S \subseteq S\]
Proof
TODO