Interior Points, Boundary Points, and Boundary of a Smooth Manifold with Boundary

Let $M$ be a smooth manifold with boundary of dimension $n$ in the sense of Smooth Atlas and Smooth Manifold with Boundary, and let $p\in M$. We say that $p$ is an interior point of $M$ if there exists a chart $(U,\varphi)$ in the chosen atlas with $p\in U$ and

$$\varphi(p)\in \operatorname{int}(H^n),$$

where $H^n$ is the half-space from Closed Upper Half-Space in Euclidean Space. We say that $p$ is a boundary point of $M$ if there exists a chart $(U,\varphi)$ in the chosen atlas with $p\in U$ and

$$\varphi(p)\in \partial H^n.$$

The set of all boundary points of $M$ is denoted by

$$\partial M$$

and is called the boundary of $M$. The set of all interior points is denoted by

$$\operatorname{int}(M).$$