definition of section along subset of codomain of continuous surjection
Topics
About: topological space
The table of contents of this article
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.
Target Context
- The reader will have a definition of section along subset of codomain of continuous surjection.
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.
Main Body
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
\(*s\): \(: S \subseteq T_2 \to T_1\), \(\in \{\text{ the continuous maps }\}\)
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Conditions:
\(\pi \circ s: S \to S = id\)
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\(s\) is called "section along \(S\) of \(\pi\)".
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: \(\pi \vert_{\pi^{-1} (S)}: \pi^{-1} (S) \subseteq T_1 \to S\) 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 \(\pi \vert_{\pi^{-1} (S)}\) continuous, how is \(\pi\)'s being continuous relevant?
We mentioned that because what if \((E, M, \pi)\) is a \(C^\infty\) 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 \(C^\infty\) vectors bundle, \((E \vert_{S}, S, \pi \vert_{\pi^{-1} (S)})\), and a section of it, \(s: S \to E \vert_{S}\), we can just call it "section of \((E \vert_{S}, S, \pi \vert_{\pi^{-1} (S)})\)" not "section along \(S\) of \((E, M, \pi)\)".
When \(S\) is an embedded submanifold with boundary, \(s\) can be called "section of \((E \vert_{S}, S, \pi \vert_{\pi^{-1} (S)})\)" 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 "\(C^\infty\) 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 \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
When \(S\) is an embedded submanifold with boundary, \(C^\infty\)-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 \(C^\infty\) manifolds with boundary, \(C^k\)-ness does not change when the domain or the codomain is regarded to be the subset.
