A definition of chart on topological manifold with boundary
Topics
About: topological manifold
The table of contents of this article
Starting Context
- The reader knows a definition of topological manifold with boundary.
- The reader knows a definition of homeomorphism.
Target Context
- The reader will have a definition of chart on topological manifold with boundary.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
For any topological manifold with boundary, \(M\), the pair of any open subset, \(U \subseteq M\), and any homeomorphism, \(\phi: U \to \phi (U) \subseteq \mathbb{H}^n \text{ or } \mathbb{R}^n\), where \(\phi (U)\) is any open subset of \(\mathbb{H}^n \text{ or } \mathbb{R}^n\), denoted as \((U \subseteq M, \phi)\)
2: Note
Logically speaking, \(\phi (U)\) can be allowed only an open subset of \(\mathbb{H}^n\), because for any open subset of \(\mathbb{R}^d\), we can instead take an open subset of \(\mathbb{H}^d\) homeomorphic to it.
We allow also open subsets of \(\mathbb{R}^d\) just for convenience: for a topological manifold (without boundary), an open subset of \(\mathbb{R}^d\) is typically taken centered at the origin, and the argument would have to be changed (although it would be just a matter of translating the open subset into \(\mathbb{H}^d\) or something) for a topological manifold with boundary if the open subset of \(\mathbb{R}^d\) was not allowed: so, it is convenient for reusing arguments on topological manifold (without boundary) for topological manifold with boundary.
A topological manifold with boundary may be a topological manifold (without boundary), which is in fact a topological manifold with empty boundary, and in that case, each \(\phi (U)\) becomes a open subset of \(\mathbb{R}^n\).
