description/proof of that for \(C^\infty\) manifold with boundary, interior point has chart ball and boundary point has chart half ball
Topics
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of chart ball around point on \(C^\infty\) manifold with boundary.
- The reader knows a definition of chart half ball around point on \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary, each interior point has a chart ball and each boundary point has a chart half ball.
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:
\(M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\(p\): \(\in M\)
//
Statements:
(
\(p \in \{\text{ the interior points of } M\}\)
\(\implies\)
\(\exists B_p \in \{\text{ the chart balls around } p \text{ on } M\}\)
)
\(\land\)
(
\(p \in \{\text{ the boundary points of } M\}\)
\(\implies\)
\(\exists H_p \in \{\text{ the chart half balls around } p \text{ on } M\}\)
)
//
2: Natural Language Description
For any \(C^\infty\) manifold with boundary, \(M\), and any \(p \in M\), if \(p\) is any interior point of \(M\), there is a chart ball around \(p\), \(B_p \subseteq M\), and if \(p\) is any boundary point of \(M\), there is a chart half ball around \(p\), \(H_p \subseteq M\).
3: Proof
Whole Strategy: Step 1: suppose that \(p\) is any interior point and take any chart around \(p\), \((U_p \subseteq M, \phi_p)\); Step 2: take an open ball around \(\phi_p (p)\) contained in the chart range; Step 3: take the preimage of the open ball under the chart map and see that it is a chart ball; Step 4: suppose that \(p\) is any boundary point and take any chart around \(p\), \((U_p \subseteq M, \phi_p)\); Step 5: take an open half ball around \(\phi_p (p)\) contained in the chart range; Step 6: take the preimage of the open half ball under the chart map and see that it is a chart half ball.
Step 1:
Let us suppose that \(p\) is any interior point.
Let us take any chart around \(p\), \((U_p \subseteq M, \phi_p)\), such that the range, \(\phi_p (U_p) \subseteq \mathbb{R}^d\), is an open subset of \(\mathbb{R}^d\): the range of a chart may be an open subset of \(\mathbb{H}^d\), but when \(p\) is any interior point, there is a chart whose range is an open subset of \(\mathbb{R}^d\).
Step 2:
There is an open ball around \(\phi_p (p)\), \(B_{\phi_p (p), \epsilon} \subseteq \mathbb{R}^d\) such that \(B_{\phi_p (p), \epsilon} \subseteq \phi_p (U_p)\), by the definition of Euclidean topology. \(B_{\phi_p (p), \epsilon}\) is open on \(\phi_p (U_p)\), by the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
Step 3:
Let us take the preimage, \(B_p := {\phi_p}^{-1} (B_{\phi_p (p), \epsilon}) \subseteq U_p\). \(B_p\) is an open neighborhood of \(p\) on \(U_p\), because \(\phi_p\) is homeomorphic. \(B_p\) is an open neighborhood of \(p\) on \(M\), by the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
\((B_p \subseteq M, \phi_p \vert_{B_p})\) is a chart, because \(\phi_p \vert_{B_p}: B_p \to B_{\phi_p (p), \epsilon}\) is obviously a homeomorphism. \(B_{\phi_p (p), \epsilon}\) is an open ball. So, \(B_p\) is an chart ball around \(p\).
Step 4:
Let us suppose that \(p\) is any boundary point.
Let us take any chart around \(p\), \((U_p \subseteq M, \phi_p)\). The range, \(\phi_p (U_p) \subseteq \mathbb{H}^d\), is an open subset of \(\mathbb{H}^d\), where \(\phi_p (p)\) is on the boundary of \(\mathbb{H}^d\).
\(\phi_p (U_p) = U' \cap \mathbb{H}^d\) for an open subset, \(U' \in \mathbb{R}^d\), by the definition of subspace topology, while \(\mathbb{H}^d\) is a subspace of \(\mathbb{R}^d\). There is an open ball around \(\phi_p (p)\), \(B_{\phi_p (p), \epsilon} \subseteq \mathbb{R}^d\), such that \(B_{\phi_p (p), \epsilon} \subseteq U'\), by the definition of Euclidean topology. As \(\phi_p (p)\) is on the boundary of \(\mathbb{H}^d\), \(H_{\phi_p (p), \epsilon} := B_{\phi_p (p), \epsilon} \cap \mathbb{H}^d\) is an open half ball, and \(H_{\phi_p (p), \epsilon} \subseteq \phi_p (U_p)\). \(H_{\phi_p (p), \epsilon}\) is open on \(\phi_p (U_p)\), by the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
Step 6:
Let us take the preimage, \(H_p := \phi_p^{-1} (H_{\phi_p (p), \epsilon})\), which is open on \(U_p\), because \(\phi_p\) is homeomorphic, and is open on \(M\), by the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
\((H_p \subseteq M, \phi_p \vert_{H_p})\) is a chart, because \(\phi_p \vert_{H_p}: H_p \to H_{\phi_p (p), \epsilon}\) is homeomorphic. \(H_{\phi_p (p), \epsilon}\) is an open half ball. So, \(H_p\) is a chart half ball around \(p\).
