A \(n\)-manifold \(M\) is called manifold with boundary if for every point there is a local chart \((U,\varphi)\) with either \(\varphi(U)\subseteq \mathbb{R}^n\) is open or \(\varphi(U)\subseteq \mathbb{H}^n=\{x\in \mathbb{R}^n\mid x_n\ge 0\}\) is open (regarding the subspace topology) with \(\varphi(U)\cap \mathbb{H}^n\neq \emptyset\). The former chart is called interior chart and the latter boundary chart. A point \(p \in M\) is called boundary point if it is in the domain of a boundary chart \(\varphi\) which sends \(p\) to \(\partial \mathbb{H}^n=\{x\in \mathbb{R}^n\mid x_n=0\}\), otherwise it is called interior point.
The set of all boundary points is denoted by \(\partial M\) and set of all interior points by \(\Int M\).
Remarks