382: Continuous Embedding

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A definition of continuous embedding

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of continuous map.
• The reader knows a definition of injection.
• The reader knows a definition of homeomorphism.

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Target Context

• The reader will have a definition of continuous embedding.

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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.

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Main Body

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1: Definition

For any topological spaces, $T_1,T_2$, any continuous map, $f:T_1\to T_2$, such that $f$ is an injection and the codomain restriction, $f^{\prime }:T_1\to f(T_1)$ is a homeomorphism

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2: Note

'Continuous embedding' and '$C^{\mathrm{\infty }}$ embedding' are different, while any $C^{\mathrm{\infty }}$ embedding is a continuous embedding, a continuous embedding is not necessarily a $C^{\mathrm{\infty }}$ embedding.

Usually just 'embedding' is used as it is usually obvious which: non-$C^{\mathrm{\infty }}$ map cannot be a $C^{\mathrm{\infty }}$ embedding and (just) embedding-ness of a $C^{\mathrm{\infty }}$ map is customarily understood to be $C^{\mathrm{\infty }}$ embedding-ness.

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References

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