1158: Subset of C∞ Manifold with Boundary That Satisfies Local-Slice Condition

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definition of subset of $C^{\mathrm{\infty }}$ manifold with boundary that satisfies local-slice condition

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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

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Starting Context

• The reader knows a definition of J-slice of chart domain with respect to point.

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Target Context

• The reader will have a definition of subset of $C^{\mathrm{\infty }}$ manifold with boundary that satisfies local-slice condition.

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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:
$M$: $\in \{\text{ the }{d}^{\prime }\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$\ast S$: $\subseteq M$
$d$: $\in \mathbb{N}$ such that $d\le d^{\prime }$
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Conditions:
$\mathrm{\forall }s\in S(\mathrm{\exists }(U_s\subseteq M,{\varphi }_s)\in \{\text{ the charts around }s\text{ for }M\},\mathrm{\exists }J\subseteq \{1,...,d^{\prime }\}=(j_1,...,j_d),\mathrm{\exists }u\in U_s(U_s\cap S=S_{J,u}(U_s)))$
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2: Note

$(U_s\subseteq M,{\varphi }_s)$ is called "adopted chart".

$(U_s\cap S\subseteq S,{\pi }_J\circ {\varphi }_s{|}_{U_s\cap S})$ is called "corresponding adopting chart".

$\{(U_s\cap S\subseteq S,{\pi }_J\circ {\varphi }_s{|}_{U_s\cap S})|s\in S\}$ is called "adopting atlas for $S$".

$S$ becomes an embedded submanifold with boundary of $M$ with the subspace topology and the adopting atlas, by the proposition that any subset of any $C^{\mathrm{\infty }}$ manifold with boundary that satisfies the local-slice condition is an embedded submanifold with boundary of the manifold with boundary with the subspace topology and the adopting atlas.

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References

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