description/proof of that for compact \(C^\infty\) manifold with boundary, sequence of points has convergent subsequence
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of compact topological space.
- The reader knows a definition of convergence of net with directed index set.
- The reader admits the proposition that any 2nd-countable topological space is 1st-countable.
- The reader admits the proposition that any 1st-countable topological space is sequentially compact if the space is countably compact.
Target Context
- The reader will have a description and a proof of the proposition that for any compact \(C^\infty\) manifold with boundary, any sequence of points has a convergent subsequence.
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:
\(M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\} \cap \{\text{ the compact topological spaces }\}\)
\(s\): \(\mathbb{N} \to M\)
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Statements:
\(\exists s^` \in \{\text{ the subsequences of } s\}, \exists m \in M (s^` \text{ converges to } m)\)
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2: Proof
Whole Strategy: Step 1: see that \(M\) is a 1st-countable topological space; Step 2: apply the proposition that any 1st-countable topological space is sequentially compact if the space is countably compact.
Step 1:
\(M\) is a 2nd-countable topological space, by the definition of \(C^\infty\) manifold with boundary.
\(M\) is a 1st-countable topological space, by the proposition that any 2nd-countable topological space is 1st-countable.
Step 2:
\(M\) is countably compact, because \(M\) is compact, obviously.
\(M\) is sequentially compact, by the proposition that any 1st-countable topological space is sequentially compact if the space is countably compact.
That means that any sequence has a convergent subsequence.
