description/proof of that for \(C^\infty\) embedding, range of embedding with topology and atlas induced by embedding is embedded submanifold with boundary of codomain
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) embedding.
- The reader knows a definition of embedded submanifold with boundary of \(C^\infty\) manifold with boundary.
- The reader admits the proposition that the composition of any \(C^\infty\) embedding after any diffeomorphism or any diffeomorphism after any \(C^\infty\) embedding is a \(C^\infty\) embedding.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) embedding, the range of the embedding with the topology and the atlas induced by the embedding is an embedded submanifold with boundary of the codomain.
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:
\(M_1\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ embeddings } \}\)
\(M_3\): \(= f (M_1)\) with the topology and the atlas induced by \(f\)
\(f'\): \(: M_1 \to f (M_1) \subseteq M_2\), \(= \text{ the codomain restriction of } f\)
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Statements:
\(M_3 \in \{\text{ the embedded submanifolds with boundary of } M_2\}\)
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2: Note
"the topology and the atlas induced by \(f\)" means that any subset, \(S \subseteq M_3\), is open if and only if \(f'^{-1} (S) \subseteq M_1\) is open, and any \((U \subseteq M_3, \phi)\) is a chart if and only if \((f'^{-1} (U) \subseteq M_1, \phi \circ f')\) is a chart.
This proposition is intuitively immediately guessed, but let us be at ease with conscience by rigorously proving it once and for all.
3: Proof
Whole Strategy: Step 1: see that \(M_3\) is a \(C^\infty\) manifold with boundary and let \(g: M_1 \to M_3\) be the diffeomorphism; Step 2: see that \(M_3\) has the subspace topology of \(M_2\); Step 3: see that the inclusion, \(\iota: M_3 \to M_2\) is a \(C^\infty\) embedding.
Step 1:
\(M_3\) is a \(C^\infty\) manifold with boundary, because it is just \(M_1\) with the points set replaced.
Let \(g: M_1 \to M_3\) be \(f\) with the codomain replaced.
\(g\) is a diffeomorphism, because \(M_1\) and \(M_3\) have the corresponding topologies and the corresponding atlases with just different points sets and \(g\) maps each point to the corresponding point: in other words, \(M_3\) is defined to make \(d\) diffeomorphic.
Step 2:
Let us see that \(M_3\) has the subspace topology of \(M_2\).
In fact, let us see that \(M_3\) and \(f (M_1) \subseteq M_2\) as the topological subspace have the same topology.
Let \(U \subseteq M_3\) be any open subset of \(M_3\).
\(U = f \circ g^{-1} (U)\), but \(g^{-1} (U)\) is open on \(M_1\) and \(f \circ g^{-1} (U)\) is open on \(f (M_1) \subseteq M_2\), by the definition of \(C^\infty\) embedding.
Let \(U \subseteq f (M_1) \subseteq M_2\) be any open subset of \(f (M_1) \subseteq M_2\).
\(U = g \circ f'^{-1} (U)\), but \(f'^{-1} (U)\) is open on \(M_1\), by the definition of \(C^\infty\) embedding, and \(g \circ f'^{-1} (U)\) is open on \(M_3\).
So, \(M_3\) and \(f (M_1) \subseteq M_2\) have the same topology.
As \(f (M_1)\) has the subspace topology of \(M_2\), \(M_3\) has the subspace topology of \(M_2\).
Step 3:
Let \(\iota: M_3 \to M_2\) be the inclusion.
\(\iota = f \circ g^{-1}\), which is a \(C^\infty\) embedding, by the proposition that the composition of any \(C^\infty\) embedding after any diffeomorphism or any diffeomorphism after any \(C^\infty\) embedding is a \(C^\infty\) embedding.
