162: C∞ Embedding

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definition of $C^{\mathrm{\infty }}$ embedding

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.
• The reader knows a definition of injection.
• The reader knows a definition of $C^{\mathrm{\infty }}$ immersion.
• The reader knows a definition of homeomorphism.

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Target Context

• The reader will have a definition of $C^{\mathrm{\infty }}$ 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: Structured Description

Here is the rules of Structured Description.

Entities:
$M_1$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$\ast f$: $:M_1\to M_2$
$f^{\prime }$: $:M_1\to f(M_1)\subseteq M_2$, $=\text{ the codomain restriction of }f$
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Conditions:
$f\in \{\text{ the injections }\}\cap \{\text{ the }{C}^{\mathrm{\infty }}\text{ immersions }\}$
$\wedge$
$f^{\prime }\in \{\text{ the homeomorphisms }\}$
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2: Natural Language Description

For any $C^{\mathrm{\infty }}$ manifolds with boundary, $M_1,M_2$, any map, $f:M_1\to M_2$, such that $f$ is an injective $C^{\mathrm{\infty }}$ immersion and the codomain restriction, $f^{\prime }:M_1\to f(M_1)\subseteq M_2$ is a homeomorphism

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3: 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.

Often just 'embedding' is used as it is often 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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