Let $M$ be a smooth manifold with boundary of dimension $n$ in the sense of \ref{def:smooth-manifold-with-boundary-2026a}, 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

where $H^n$ is the half-space from \ref{def:closed-upper-half-space-euclidean-2026a}. 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

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

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