Let \(X\) be a topological space and \(A\subseteq X\). The boundary of \(A\) is \(\bd A \coloneqq \cl_X(A) \sm A^\circ\), where \(\cl_X(A)\) is the closure of \(A\) and \(A^\circ\), its interior, is the set of interior points of \(A\).
A point \(p\in X\) lies in \(\bd A\) if and only if every neighborhood of \(p\) intersects both \(A\) and \(X\sm A\).