description/proof of that for \(C^\infty\) manifold with boundary, immersed submanifold with boundary, and open submanifold with boundary, intersection of immersed submanifold with boundary and open submanifold with boundary is immersed submanifold with boundary of open submanifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of immersed submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader knows a definition of open submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any immersed submanifold with boundary of any \(C^\infty\) manifold with boundary, any subset that is open by the subspace topology is open, and the immersed submanifold with boundary is an embedded submanifold with boundary if and only if each open subset is open by the subspace topology.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary, any immersed submanifold with boundary of the manifold with boundary, and any open submanifold with boundary of the manifold with boundary, the intersection of the immersed submanifold with boundary and the open submanifold with boundary is an immersed submanifold with boundary of the open submanifold with boundary.
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:
\(M'\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M\): \(\in \{\text{ the immersed submanifolds with boundary of } M'\}\)
\(U'\): \(\in \{\text{ the open submanifolds with boundary of } M'\}\)
\(M \cap U'\):
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Statements:
\(M \cap U' \in \{\text{ the open submanifolds with boundary of } M\}\)
\(\land\)
\(M \cap U' \in \{\text{ the immersed submanifolds with boundary of } U'\}\)
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2: Proof
Whole Strategy: Step 1: see that \(M \cap U'\) is an open submanifold with boundary of \(M\); Step 2: see that \(M \cap U'\) is an immersed submanifold with boundary of \(U'\).
Step 1:
\(M \cap U'\) is an open subset of \(M\), by the proposition that for any immersed submanifold with boundary of any \(C^\infty\) manifold with boundary, any subset that is open by the subspace topology is open, and the immersed submanifold with boundary is an embedded submanifold with boundary if and only if each open subset is open by the subspace topology.
So, \(M \cap U'\) can be regarded to be the open submanifold with boundary of \(M\).
Step 2:
Let us think of the inclusion, \(\iota: M \to M'\), which is a \(C^\infty\) immersion, by the definition of immersed submanifold with boundary.
Let us think of its restriction, \(\iota \vert_{M \cap U'}: M \cap U' \to U'\), whose domain is an open subset of \(M\) and whose codomain is an open subset of \(M'\).
\(\iota \vert_{M \cap U'}\) is a \(C^\infty\) immersion, because \(\iota\) is so, which implies that \(\iota\) is locally so.
As \(\iota \vert_{M \cap U'}\) is the inclusion of \(M \cap U'\) into \(U'\), \(M \cap U'\) is an immersed submanifold with boundary of \(U'\).
