828: Open Subset of C∞ Trivializing Open Subset Is C∞ Trivializing Open Subset

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description/proof of that open subset of $C^{\mathrm{\infty }}$ trivializing open subset is $C^{\mathrm{\infty }}$ trivializing open subset

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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 $C^{\mathrm{\infty }}$ trivializing open subset and $C^{\mathrm{\infty }}$ local trivialization.
• The reader admits the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
• The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
• The reader admits the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction on any domain that contains the point is $C^k$ at the point.
• The reader admits the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point.

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Target Context

• The reader will have a description and a proof of the proposition that any open subset of any $C^{\mathrm{\infty }}$ trivializing open subset is a $C^{\mathrm{\infty }}$ trivializing open subset.

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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:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$E$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$k$: $\in \mathbb{N}\setminus \{0\}$
$\pi$: $:E\to M$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
$U$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ trivializing open subsets of }M\}$
$\mathrm{\Phi }$: $:{\pi }^{-1}(U)\to U\times {\mathbb{R}}^k$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ local trivializations }\}$
$V$: $\in \{\text{ the open subsets of }U\}$
$\mathrm{\Phi }{|}_{{\pi }^{-1}}(V)$: $:{\pi }^{-1}(V)\to V\times {\mathbb{R}}^k$
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Statements:
$V\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ trivializing open subsets of }M\}$
$\wedge$
$\mathrm{\Phi }{|}_{{\pi }^{-1}}(V)\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ local trivializations }\}$
//

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2: Proof

Whole Strategy: Step 1: see that $V$ is an open subset of $M$; Step 2: see that $\mathrm{\Phi }{|}_{{\pi }^{-1}}(V)$ is indeed $:{\pi }^{-1}(V)\to V\times {\mathbb{R}}^k$; Step 3: see that $\mathrm{\Phi }{|}_{{\pi }^{-1}}(V)$ is a homeomorphism; Step 4: see that for each $m\in V$, $\mathrm{\Phi }{|}_{{\pi }^{-1}}(V){|}_{{\pi }^{-1}(m)}:{\pi }^{-1}(m)\to \{m\}\times {\mathbb{R}}^k$ is a 'vectors spaces - linear morphisms' isomorphism; Step 5: see that $\mathrm{\Phi }{|}_{{\pi }^{-1}}(V)$ is a diffeomorphism.

Step 1:

$V$ is an open subset of $M$, by the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.

Step 2:

$\mathrm{\Phi }{|}_{{\pi }^{-1}}(V)$ is indeed $:{\pi }^{-1}(V)\to V\times {\mathbb{R}}^k$, because for each $m\in U$, $\mathrm{\Phi }{|}_{{\pi }^{-1}}(m)$ is 'vectors spaces - linear morphisms' isomorphic $:{\pi }^{-1}(m)\to \{m\}\times {\mathbb{R}}^k$.

Step 3:

$\mathrm{\Phi }{|}_{{\pi }^{-1}}(V)$ is a homeomorphism, because it is the domain and the codomain restriction of homeomorphic $\mathrm{\Phi }$, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.

Step 4:

For each $m\in V$, $\mathrm{\Phi }{|}_{{\pi }^{-1}}(V){|}_{{\pi }^{-1}(m)}:{\pi }^{-1}(m)\to \{m\}\times {\mathbb{R}}^k$ is a 'vectors spaces - linear morphisms' isomorphism, because it equals $\mathrm{\Phi }{|}_{{\pi }^{-1}(m)}$, which is a 'vectors spaces - linear morphisms' isomorphism.

Step 5:

$\mathrm{\Phi }{|}_{{\pi }^{-1}}(V)$ is a diffeomorphism, because it is the domain and the codomain restriction of $\mathrm{\Phi }$, by the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction on any domain that contains the point is $C^k$ at the point and the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point.

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References

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