Manifold
Manifold with boundary

A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an n {\displaystyle n} -manifold with boundary is an (n − 1) {\displaystyle (n-1)} -manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A square with interior is also a 2-manifold with boundary. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. (Do not confuse with Boundary (topology)).

In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open n {\displaystyle n} -ball { (x 1, x 2, …, x n) | Σ x i 2 < 1 } {\displaystyle \{(x_{1},x_{2},\dots,x_{n})\vert \Sigma x_{i}^{2}<1\}}. Every boundary point has a neighborhood homeomorphic to the "half" n {\displaystyle n} -ball { (x 1, x 2, …, x n) | Σ x i 2 < 1 and x 1 ≥ 0 } {\displaystyle \{(x_{1},x_{2},\dots,x_{n})\vert \Sigma x_{i}^{2}<1{\text{ and }}x_{1}\geq 0\}}. Any homeomorphism between half-balls must send points with x 1 = 0 {\displaystyle x_{1}=0} to points with x 1 = 0 {\displaystyle x_{1}=0}. This invariance allows to "define" boundary points; see next paragraph.
keywords:existence