description/proof of that for immersed submanifold with boundary of \(C^\infty\) manifold with boundary, subset that is open by subspace topology is open, and it is embedded submanifold with boundary iff open subset is open by subspace topology
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 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.
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:
\(M'\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary } \}\)
\(M\): \(\in \{\text{ the immersed submanifolds with boundary of } M'\}\)
//
Statements:
\(\forall U' \in \{\text{ the open subsets of } M'\} (U' \cap M \in \{\text{ the open subsets of } M\})\)
\(\land\)
(
\(\forall U \in \{\text{ the open subsets of } M\} (\exists U' \in \{\text{ the open subsets of } M'\} (U = U' \cap M))\)
\(\iff\)
\(M \in \{\text{ the embedded submanifolds with boundary of } M'\}\)
)
//
2: Note
For example, the image of the figure-8 curve, \(f: (- \pi, \pi) \to M' = \mathbb{R}^2, t \mapsto (sin (2 t), sin (t))\), \(M\), is a non-embedded immersed submanifold of \(M'\), and although an open neighborhood around \(f (0) = (0, 0)\) on \(M\) is not open by the subspace topology of \(M'\) for \(M\), for any open neighborhood of \((0, 0)\) on \(M'\), \(U'_{(0, 0)}\), \(U'_{(0, 0)} \cap M\) is open on \(M\).
3: Proof
Whole Strategy: Step 1: take the inclusion, \(\iota: M \to M'\), and see that \(U' \cap M = \iota^{-1} (U')\), which is open on \(M\); Step 2: suppose that \(\forall U \in \{\text{ the open subsets of } M\} (\exists U' \in \{\text{ the open subsets of } M'\} (U = U' \cap M))\); Step 3: see that \(M\) has the subspace topology of \(M'\) and so, \(\iota\) is a \(C^\infty\) embedding; Step 4: suppose that \(M\) is an embedded submanifold with boundary; Step 5: see that \(\forall U \in \{\text{ the open subsets of } M\} (\exists U' \in \{\text{ the open subsets of } M'\} (U = U' \cap M))\) holds.
Step 1:
Let \(\iota: M \to M'\) be the inclusion, which is an injective \(C^\infty\) immersion, by the definition of immersed submanifold with boundary.
Let \(U' \subseteq M'\) be any open subset of \(M'\).
\(U' \cap M = \iota^{-1} (U')\), because for each \(p \in U' \cap M\), \(\iota (p) = p \in U'\), so, \(p \in \iota^{-1} (U')\); for each \(p \in \iota^{-1} (U')\), \(\iota (p) = p \in U'\) and \(p \in M\), so, \(p \in U' \cap M\).
\(\iota\) is continuous, because it is \(C^\infty\), so, \(\iota^{-1} (U')\) is open on \(M\).
So, \(U' \cap M\) is open on \(M\).
Step 2:
Let us suppose that \(\forall U \in \{\text{ the open subsets of } M\} (\exists U' \in \{\text{ the open subsets of } M'\} (U = U' \cap M))\).
Step 3:
By the supposition and Step 1, any subset of \(M\) is open if and only if it is \(U' \cap M\), which means that \(M\) has the subspace topology of \(M'\).
So, the inclusion, \(\iota: M \to \iota (M) \subseteq M'\) is a homeomorphism.
So, \(\iota: M \to M'\) is a \(C^\infty\) embedding.
So, \(M\) is an embedded submanifold with boundary of \(M'\).
Step 4:
Let us suppose that \(M\) is an embedded submanifold with boundary of \(M'\).
Step 5:
\(M\) has the subspace topology of \(M'\), by the definition of embedded submanifold with boundary.
So, \(\forall U \in \{\text{ the open subsets of } M\} (\exists U' \in \{\text{ the open subsets of } M'\} (U = U' \cap M))\) holds.
