A definition of topological manifold with boundary
Topics
About: manifold
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of 2nd-countable topological space.
- The reader knows a definition of locally topologically closed upper half Euclidean topological space.
Target Context
- The reader will have a definition of 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
Any Hausdorff, 2nd-countable, locally topologically closed upper half Euclidean topological space
2: Note
It is important to distinguish various Euclidean-nesses, because many people use just "Euclidean" meaning also having the Euclidean metric, the Euclidean maximal atlas, and its Riemannian connection (the Euclidean connection). Here it is only topologically Euclidean.
A topological manifold with boundary may be a topological manifold (without boundary), which is in fact a topological manifold with empty boundary, because the locally topologically closed upper half Euclidean topological space may be a locally topologically Euclidean topological space.
When we use the term, 'topological manifold with boundary', it always includes 'topological manifold (without boundary)'. When we specifically mean a topological manifold with nonempty boundary, we will specifically state so.
