Boundary (of a manifold)

The subset of the closure $\overline{M^n}$ of an (open) $n$- dimensional real manifold $M^n$ for which a neighbourhood of each point is homeomorphic to some domain $W^n$ in the closed half-space of ${\mathbf{R}}^n$, the domain being open in ${\mathbf{R}}_{+}^n$( but not in ${\mathbf{R}}^n$). A point $a\in \overline{M^n}$ corresponding to a boundary point of $W^n\subset {\mathbf{R}}_{+}^n$, i.e. to an intersection point of $\overline{W^n}$ with the boundary of ${\mathbf{R}}_{+}^n$, is called a boundary point of $M^n$. A manifold having boundary points is known as a manifold with boundary. A compact manifold without boundary is known as a closed manifold. The set of all boundary points of $M^n$ is an $(n-1)$- dimensional manifold without boundary.

Comments[edit]

References[edit]