2022-10-30

382: Continuous Embedding

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A definition of continuous embedding

Topics


About: topological space

The table of contents of this article


Starting Context



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.


References


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