852: For C∞ Embedding, Range of Embedding with Topology and Atlas Induced by Embedding Is Embedded Submanifold with Boundary of Codomain

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description/proof of that for $C^{\mathrm{\infty }}$ embedding, range of embedding with topology and atlas induced by embedding is embedded submanifold with boundary of codomain

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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: Note
• 3: Proof

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ embedding.
• The reader knows a definition of embedded submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader admits the proposition that the composition of any $C^{\mathrm{\infty }}$ embedding after any diffeomorphism or any diffeomorphism after any $C^{\mathrm{\infty }}$ embedding 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 $C^{\mathrm{\infty }}$ embedding, the range of the embedding with the topology and the atlas induced by the embedding is an embedded submanifold with boundary of the codomain.

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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:
$M_1$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$f$: $:M_1\to M_2$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ embeddings }\}$
$M_3$: $=f(M_1)$ with the topology and the atlas induced by $f$
$f^{\prime }$: $:M_1\to f(M_1)\subseteq M_2$, $=\text{ the codomain restriction of }f$
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Statements:
$M_3\in \{\text{ the embedded submanifolds with boundary of }{M}_2\}$
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2: Note

"the topology and the atlas induced by $f$" means that any subset, $S\subseteq M_3$, is open if and only if $f^{\prime -1}(S)\subseteq M_1$ is open, and any $(U\subseteq M_3,\varphi )$ is a chart if and only if $(f^{\prime -1}(U)\subseteq M_1,\varphi \circ f^{\prime })$ is a chart.

This proposition is intuitively immediately guessed, but let us be at ease with conscience by rigorously proving it once and for all.

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3: Proof

Whole Strategy: Step 1: see that $M_3$ is a $C^{\mathrm{\infty }}$ manifold with boundary and let $g:M_1\to M_3$ be the diffeomorphism; Step 2: see that $M_3$ has the subspace topology of $M_2$; Step 3: see that the inclusion, $\iota :M_3\to M_2$ is a $C^{\mathrm{\infty }}$ embedding.

Step 1:

$M_3$ is a $C^{\mathrm{\infty }}$ manifold with boundary, because it is just $M_1$ with the points set replaced.

Let $g:M_1\to M_3$ be $f$ with the codomain replaced.

$g$ is a diffeomorphism, because $M_1$ and $M_3$ have the corresponding topologies and the corresponding atlases with just different points sets and $g$ maps each point to the corresponding point: in other words, $M_3$ is defined to make $d$ diffeomorphic.

Step 2:

Let us see that $M_3$ has the subspace topology of $M_2$.

In fact, let us see that $M_3$ and $f(M_1)\subseteq M_2$ as the topological subspace have the same topology.

Let $U\subseteq M_3$ be any open subset of $M_3$.

$U=f\circ g^{-1}(U)$, but $g^{-1}(U)$ is open on $M_1$ and $f\circ g^{-1}(U)$ is open on $f(M_1)\subseteq M_2$, by the definition of $C^{\mathrm{\infty }}$ embedding.

Let $U\subseteq f(M_1)\subseteq M_2$ be any open subset of $f(M_1)\subseteq M_2$.

$U=g\circ f^{\prime -1}(U)$, but $f^{\prime -1}(U)$ is open on $M_1$, by the definition of $C^{\mathrm{\infty }}$ embedding, and $g\circ f^{\prime -1}(U)$ is open on $M_3$.

So, $M_3$ and $f(M_1)\subseteq M_2$ have the same topology.

As $f(M_1)$ has the subspace topology of $M_2$, $M_3$ has the subspace topology of $M_2$.

Step 3:

Let $\iota :M_3\to M_2$ be the inclusion.

$\iota =f\circ g^{-1}$, which is a $C^{\mathrm{\infty }}$ embedding, by the proposition that the composition of any $C^{\mathrm{\infty }}$ embedding after any diffeomorphism or any diffeomorphism after any $C^{\mathrm{\infty }}$ embedding is a $C^{\mathrm{\infty }}$ embedding.

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References

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