A definition of continuous embedding
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of continuous map.
- The reader knows a definition of injection.
- The reader knows a definition of homeomorphism.
Target Context
- The reader will have a definition of continuous embedding.
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 spaces, \(T_1, T_2\), any continuous map, \(f: T_1 \rightarrow T_2\), such that \(f\) is an injection and the codomain restriction, \(f': T_1 \rightarrow f (T_1)\) is a homeomorphism
2: Note
'Continuous embedding' and '\(C^\infty\) embedding' are different, while any \(C^\infty\) embedding is a continuous embedding, a continuous embedding is not necessarily a \(C^\infty\) embedding.
Usually just 'embedding' is used as it is usually obvious which: non-\(C^\infty\) map cannot be a \(C^\infty\) embedding and (just) embedding-ness of a \(C^\infty\) map is customarily understood to be \(C^\infty\) embedding-ness.
