825: Section Along Subset of Codomain of Continuous Surjection

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definition of section along subset of codomain of continuous surjection

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Topics

About: topological space

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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 continuous map.
• The reader knows a definition of surjection.
• The reader knows a definition of subspace topology of subset of topological space.

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Target Context

• The reader will have a definition of section along subset of codomain of continuous surjection.

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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:
$T_1$: $\in \{\text{ the topological spaces }\}$
$T_2$: $\in \{\text{ the topological spaces }\}$
$\pi$: $:T_1\to T_2$, $\in \{\text{ the continuous surjections }\}$
$S$: $\subseteq T_2$, with the subspace topology
$\ast s$: $:S\subseteq T_2\to T_1$, $\in \{\text{ the continuous maps }\}$
//

Conditions:
$\pi \circ s:S\to S=id$
//

$s$ is called "section along $S$ of $\pi$".

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2: Note

Allowing $S$ to have another topology is logically possible, but does not seem so meaningful to be defined based on $\pi$, because the relation with $\pi$'s being continuous would become thin: $\pi {|}_{{\pi }^{-1}(S)}:{\pi }^{-1}(S)\subseteq T_1\to S$ with ${\pi }^{-1}(S)$ regarded to be the topological subspace of $T_1$ might not be continuous any more; if also ${\pi }^{-1}(S)$ was allowed to have another topology in order to make $\pi {|}_{{\pi }^{-1}(S)}$ continuous, how is $\pi$'s being continuous relevant?

We mentioned that because what if $(E,M,\pi )$ is a $C^{\mathrm{\infty }}$ vectors bundle and $S\subseteq M$ is an immersed submanifold with boundary? $s:S\to E$ with $S$ regarded to be an immersed submanifold with boundary is against this definition, because $S$ is not a topological subspace of $M$ in general. If we want to talk about the restricted $C^{\mathrm{\infty }}$ vectors bundle, $(E{|}_S,S,\pi {|}_{{\pi }^{-1}(S)})$, and a section of it, $s:S\to E{|}_S$, we can just call it "section of $(E{|}_S,S,\pi {|}_{{\pi }^{-1}(S)})$" not "section along $S$ of $(E,M,\pi )$".

When $S$ is an embedded submanifold with boundary, $s$ can be called "section of $(E{|}_S,S,\pi {|}_{{\pi }^{-1}(S)})$" or "section along $S$ of $(E,M,\pi )$", because $S$ has the subspace topology.

The reason why we do not call it just "section on $S$" is that when $E=TM$, it would be as though $s:S\to TS$ instead of $:S\to TM$.

When $s$ is called "$C^{\mathrm{\infty }}$ section along $S$", it is as a map from a subset of $M$ into $E$ according to the definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.

When $S$ is an embedded submanifold with boundary, $C^{\mathrm{\infty }}$-ness does not change if the domain of $s$ is regarded to be the embedded submanifold with boundary, by the proposition that for any map between any embedded submanifolds with boundary of any $C^{\mathrm{\infty }}$ manifolds with boundary, $C^k$-ness does not change when the domain or the codomain is regarded to be the subset.

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References

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