Let \(X\) be a topological space and \(A\subseteq X\). The subset \(A\) is closed if and only if it contains all of its boundary points .
Proof
This is an immediate consequence of the characterizations of boundaries
and the closure
.
Let \(X\) be a topological space and \(A\subseteq X\). The subset \(A\) is closed if and only if it contains all of its boundary points .