description/proof of that for \(C^\infty\) vectors bundle, there is chart trivializing open cover
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle of rank \(k\).
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, a trivializing open subset is not necessarily a chart open subset, but there is a possibly smaller chart trivializing open subset at each point on any trivializing open subset.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart.
- The reader admits the proposition that any open subset of any \(C^\infty\) trivializing open subset is a \(C^\infty\) trivializing open subset.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle, there is a chart trivializing open cover.
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:
\((E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
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Statements:
\(\exists \text{ a chart trivializing open cover of } M\)
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2: Proof
Whole Strategy: Step 1: take any trivializing open cover, \(\{U_\beta \vert \beta \in B\}\); Step 2: for each \(p \in U_\beta\), take a chart trivializing open subset, \(U_{\beta, p}\), and see that \(\{U_{\beta, p} \vert \beta \in B, p \in U_\beta\}\) covers \(M\); Step 3: see that \(\{U_{\beta, p} \vert \beta \in B, p \in U_\beta\}\) can be refined to be locally finite, if so desired.
Step 1:
There is a trivializing open cover, \(\{U_\beta \vert \beta \in B\}\), where \(B\) is any possibly uncountable index set, by the definition of \(C^\infty\) vectors bundle.
But \(U_\beta\) is not necessarily a chart open subset, so, we are going to find a trivializing open cover whose each constituent is a chart open subset, which this proposition is about.
Step 2:
For each \(p \in U_\beta\), there is a chart trivializing open subset, \(U_{\beta, p}\) such that \(U_{\beta, p} \subseteq U_\beta\), by the proposition that for any \(C^\infty\) vectors bundle, a trivializing open subset is not necessarily a chart open subset, but there is a possibly smaller chart trivializing open subset at each point on any trivializing open subset.
Although \(\{U_{\beta, p} \vert \beta \in B, p \in U_\beta\}\) may have some duplications, such duplications are automatically eliminated by the definition of set.
\(\{U_{\beta, p} \vert \beta \in B, p \in U_\beta\}\) obviously covers \(M\).
Step 3:
The proposition has been already proved, but if one desires the open cover to be smaller, \(\{U_{\beta, p} \vert \beta \in B, p \in U_\beta\}\) can be refined to be locally finite, because any \(C^\infty\) manifold with boundary is paracompact: each element of the refinement is a chart trivializing open subset, by the proposition that for any \(C^\infty\) manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart and the proposition that any open subset of any \(C^\infty\) trivializing open subset is a \(C^\infty\) trivializing open subset.
