Definition:Topological Manifold with Boundary
Definition
Let $\R^n$ denote the $n$-dimensional Euclidean space.
Let $\H^n$ denote the $n$-dimensional closed upper half-space of $\R^n$.
A $d$-dimensional topological manifold with boundary is a second-countable Hausdorff space $M$ in which every point has a neighborhood homeomorphic either to:
- a) an open subset of $\R^n$ or
- b) an open subset of $\H^n$.
Chart
A chart of $M$ is an ordered pair $\struct {U, \varphi}$, where:
- $U$ is an open subset of $M$
- $\varphi: U \to D \subseteq \R^n$ is a homeomorphism of $U$ onto an open subset $D$ of $\R^n$ or $\H^n$.
Interior Chart
The chart $\struct {U, \varphi}$ is an interior chart if and only if $\map \varphi U$ is an open subset of $\R^n$.
Boundary Chart
Let $\partial \H^n$ denote the boundary of the closed upper half-space.
The chart $\struct {U, \varphi}$ is a boundary chart if and only if $\map \varphi U$ is an open subset of $\H^n$ such that:
- $\map \varphi U \cap \partial \H^n \ne \O$
Coordinate Half-Ball
Let $p \in M$.
Let $r \in \R_{>0}$ be a strictly positive real number.
Let $\map {B_r} p$ denote an open ball of radius $r$ centered on $p$.
A coordinate half-ball is a chart $\struct {U, \varphi}$ such that:
- $\exists x \in \partial \H^n : \varphi \sqbrk U = \map {B_r} x \cap \H_n$
Interior and Boundary
Interior Point
A point $p \in M$ is an interior point of $M$ if and only if $p$ is in the domain of some interior chart of $M$.
Interior
The interior of $M$, denoted $\Int M$, is the set of all its interior points.
Boundary Point
A point $p \in M$ is a boundary point of $M$ if and only if $p$ is in the domain of some boundary chart of $M$ that send $p$ to $\partial \H^n$.
Boundary
The boundary of $M$, denoted $\partial M$, is the set of all its boundary points.
Also see
Sources
- 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): Chapter $1$: Smooth Manifolds: $\S$ Manifolds with Boundary