Let \(n\geq 1\) and let \(M\) be a topological $n$-manifold with boundary. The boundary of \(M\) is the subspace \[ \del M \coloneqq \theset{x\in M \st H_n(M, M\setminus\theset{x}; \ZZ) = 0} \] [@Hat02].
A point \(x\in M\) lies in \(\del M\) if and only if a neighborhood of \(x\) is carried homeomorphically onto an open subset of \(\theset{y\in\RR^n \st y_n\geq 0}\) with \(x\) going to a point with \(y_n = 0\); for every other point, \(H_n(M, M\setminus\theset{x};\ZZ)\cong\ZZ\). The boundary \(\del M\) is an \((n-1)\)-manifold with empty boundary [@Hat02].