367: For C^infty Manifold and Its Regular Submanifold, Open Subset of Super Manifold Is C^\infty Manifold and Intersection of Open Subset and Regular Submanifold Is Regular Submanifold of Open Subset Manifold

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A description/proof of that for $C^{\mathrm{\infty }}$ manifold and its regular submanifold, open subset of super manifold is $C^{\mathrm{\infty }}$ manifold and intersection of open subset and regular submanifold is regular submanifold of open subset manifold

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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: Description
• 2: Proof

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Starting Context

• The reader knows a definition of regular submanifold.
• The reader admits the proposition that for any topological space, the intersection of any basis and any subspace is a basis for the subspace.

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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 and its any regular submanifold, any open subset of the super manifold is canonically a $C^{\mathrm{\infty }}$ manifold, and the intersection of the open subset and the regular submanifold is a regular submanifold of the open subset manifold.

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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: Description

For any $C^{\mathrm{\infty }}$ manifold, $M^{\prime }$, and any regular submanifold, $M\subseteq M^{\prime }$, any open subset, $U^{\prime }\subseteq M^{\prime }$, is a $C^{\mathrm{\infty }}$ manifold with the subspace topology and the restricted atlas, and $U^{\prime }\cap M$ is a regular submanifold of $U^{\prime }$.

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2: Proof

$U^{\prime }$ is Hausdorff as a subspace of the Hausdorff space, is 2nd countable, because the basis for $U^{\prime }$ is the intersection of the countable basis for $M^{\prime }$ and $U^{\prime }$, by the proposition that for any topological space, the intersection of any basis and any subspace is a basis for the subspace, is locally Euclidean, because around any point, $p^{\prime }\in U^{\prime }$, there is an open neighborhood on $M^{\prime }$ contained in $U^{\prime }$. The restriction of the atlas for $M^{\prime }$ is a $C^{\mathrm{\infty }}$ atlas for $U^{\prime }$, because the intersection of any chart for $M^{\prime }$ and $U^{\prime }$ is a chart for $U^{\prime }$, the transition functions are $C^{\mathrm{\infty }}$, and the charts cover $U^{\prime }$. So, $U^{\prime }$ is a $C^{\mathrm{\infty }}$ manifold.

Around any point, $p\in U^{\prime }\cap M$, there is an adopted chart, $(U_p^{\prime }\subseteq U^{\prime }\subseteq M^{\prime },{\varphi }_p^{\prime })$, and $(U_p^{\prime }\cap M,{\varphi }_p^{\prime }{|}_{U_p^{\prime }\cap M})$ is the corresponding adopting chart. $(U_p^{\prime },{\varphi }_p^{\prime })$ can be taken to be an adopted chart also for $U^{\prime }\cap M$, because $U_p^{\prime }\cap (U^{\prime }\cap M)=U_p^{\prime }\cap M$, which equals the subset of $U_p^{\prime }$ whose image under ${\varphi }_p^{\prime }$ is a slice of ${\varphi }_p^{\prime }(U_p^{\prime })$.

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References

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