Topological Boundary#
Definition: Boundary Point
Let \(S \subseteq X\) be a subset of a topological space.
We say that \(p \in X\) is a boundary point of \(S\) if every neighborhood of \(p\) intersects both \(S\) and its complement \(X \setminus S\):
\[\forall N(p): N(p) \cap S \ne \varnothing \text{ and } N(p) \cap (X \setminus S) \ne \varnothing\]
Definition: Topological Boundary
The (topological) boundary of \(S\) is the set of all its boundary points.
Notation
\[\partial_X S \qquad \partial S \qquad \operatorname{Bd}_X(S) \qquad \operatorname{Bd}(S)\]