822: For C∞ Manifold with Boundary, Embedded Submanifold with Boundary of Embedded Submanifold with Boundary Is Embedded Submanifold with Boundary of Manifold with Boundary

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description/proof of that for $C^{\mathrm{\infty }}$ manifold with boundary, embedded submanifold with boundary of embedded submanifold with boundary is embedded submanifold with boundary of manifold with boundary

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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: Proof

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of embedded submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader admits the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ embedding between any $C^{\mathrm{\infty }}$ manifolds with boundary, the restriction of the embedding on any embedded submanifold with boundary domain is a $C^{\mathrm{\infty }}$ embedding.

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Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, any embedded submanifold with boundary of any embedded submanifold with boundary is an embedded submanifold with boundary of the manifold with boundary.

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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:
$M^{″}$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M^{\prime }$: $\in \{\text{ the embedded submanifolds with boundary of }{M}^{″}\}$
$M$: $\in \{\text{ the embedded submanifolds with boundary of }{M}^{\prime }\}$
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Statements:
$M$: $\in \{\text{ the embedded submanifolds with boundary of }{M}^{″}\}$
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2: Proof

Whole Strategy: Step 1: see that $M$ is a topological subspace of $M^{″}$; Step 2: let $\iota :M\to M^{\prime }$ and ${\iota }^{\prime }:M^{\prime }\to M^{″}$ be the inclusions, and see that ${\iota }^{\prime }\circ \iota :M\to M^{″}$ is a $C^{\mathrm{\infty }}$ embedding; Step 3: conclude the proposition.

Step 1:

Let us see that $M$ is a topological subspace of $M^{″}$.

$M$ is a topological subspace of $M^{\prime }$. $M^{\prime }$ is a topological subspace of $M^{″}$.

Each subset of $M$ is open or not open if and only if it is open or not open respectively with $M$ regarded as the topological subspace of $M^{″}$, by the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace, which means that $M$ is indeed a topological subspace of $M^{″}$.

Step 2:

Let $\iota :M\to M^{\prime }$ and ${\iota }^{\prime }:M^{\prime }\to M^{″}$ be the inclusions, which are $C^{\mathrm{\infty }}$ embeddings.

${\iota }^{\prime }\circ \iota :M\to M^{″}$ is the inclusion.

${\iota }^{\prime }\circ \iota$ is a $C^{\mathrm{\infty }}$ embedding, by the proposition that for any $C^{\mathrm{\infty }}$ embedding between any $C^{\mathrm{\infty }}$ manifolds with boundary, the restriction of the embedding on any embedded submanifold with boundary domain is a $C^{\mathrm{\infty }}$ embedding.

Step 3:

So, $M$ is an embedded submanifold with boundary of $M^{″}$.

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References

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