Accumulation Points#
Definition: Accumulation Point
Let \(X\) be a topological space and let \(S\) be a subset of \(X\).
We say that \(p \in X\) is an accumulation point / limit point / cluster point of \(S\) if every neighborhood \(N\) of \(p\) contains at least one point of \(S\) which is different from \(p\):
\[\exists p' \in S \text{ with } p' \in N \text{ and } p' \ne p\]
Theorem: Closedness and Accumulation Points
Let \((X, \tau)\) be a topological space.
A subset \(S \subseteq X\) is closed if and only if it contains all of its accumulation points.
Proof
TODO