A definition of locally topologically Euclidean topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of neighborhood of point.
- The reader knows a definition of homeomorphism.
- The reader knows a definition of Euclidean topological space.
Target Context
- The reader will have a definition of locally topologically Euclidean topological space.
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 Euclidean topological space, \(\mathbb{R}^n\), any topological space, \(T\), such that at its any point, \(p \in T\), there are an open neighborhood, \(U_p \subseteq T\), of \(p\), and an open neighborhood, \(U_q \subseteq \mathbb{R}^n\), of a point, \(q \in \mathbb{R}^n\), such that there is a homeomorphism, \(f: U_p \to U_q\)
2: Note
The expression, "topologically Euclidean topological space", may seem redundant, but is not so strictly speaking, because 'Riemannianly Euclidean topological space' is possible because any Riemannian manifold is a topological space, as well as 'only topologically Euclidean Riemannian manifold' is of course possible.
