A description/proof of that for \(C^\infty\) manifold and its regular submanifold, open subset of super manifold is \(C^\infty\) manifold and intersection of open subset and regular submanifold is regular submanifold of open subset manifold
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold and its any regular submanifold, any open subset of the super manifold is canonically a \(C^\infty\) manifold, and the intersection of the open subset and the regular submanifold is a regular submanifold of the open subset manifold.
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: Description
For any \(C^\infty\) manifold, \(M'\), and any regular submanifold, \(M \subseteq M'\), any open subset, \(U' \subseteq M'\), is a \(C^\infty\) manifold with the subspace topology and the restricted atlas, and \(U' \cap M\) is a regular submanifold of \(U'\).
2: Proof
\(U'\) is Hausdorff as a subspace of the Hausdorff space, is 2nd countable, because the basis for \(U'\) is the intersection of the countable basis for \(M'\) and \(U'\), by the proposition that for any topological space, the intersection of any basis and any subspace is a basis for the subspace, is locally Euclidean, because around any point, \(p' \in U'\), there is an open neighborhood on \(M'\) contained in \(U'\). The restriction of the atlas for \(M'\) is a \(C^\infty\) atlas for \(U'\), because the intersection of any chart for \(M'\) and \(U'\) is a chart for \(U'\), the transition functions are \(C^\infty\), and the charts cover \(U'\). So, \(U'\) is a \(C^\infty\) manifold.
Around any point, \(p \in U' \cap M\), there is an adopted chart, \((U'_p \subseteq U' \subseteq M', \phi'_p)\), and \((U'_p \cap M, \phi'_p \vert_{U'_p \cap M})\) is the corresponding adopting chart. \((U'_p, \phi'_p)\) can be taken to be an adopted chart also for \(U' \cap M\), because \(U'_p \cap (U' \cap M) = U'_p \cap M\), which equals the subset of \(U'_p\) whose image under \(\phi'_p\) is a slice of \(\phi'_p (U'_p)\).
