description/proof of that for \(C^\infty\) manifold with boundary, there is countable cover by inner charts of \(r'\)-\(r\)-open-balls charts pairs and \(r'\)-\(r\)-open-half-balls charts pairs
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(r'\)-\(r\)-open-balls charts pair around point on \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(r'\)-\(r\)-open-half-balls charts pair around point on \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary, each interior point has an \(r'\)-\(r\)-open-balls charts pair and each boundary point has an \(r'\)-\(r\)-open-half-balls charts pair for any positive \(r'\) and \(r\).
- The reader admits the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) manifold with boundary, there is a countable cover by the inner charts of some \(r'\)-\(r\)-open-balls charts pairs and some \(r'\)-\(r\)-open-half-balls charts pairs.
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:
\(M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
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Statements:
\(\exists \{((B_{m_j, r'_j} \subseteq M, \phi_{m_j}), (B_{m_j, r_j} \subseteq M, \phi_{m_j} \vert_{B_{m_j, r_j}})) \vert j \in J, ((B_{m_j, r'_j} \subseteq M, \phi_{m_j}), (B_{m_j, r_j} \subseteq M, \phi_{m_j} \vert_{B_{m_j, r_j}})) \in \{\text{ the } r' \text{ - } r \text{ -open-balls charts pairs around } m_j\}\}, \exists \{((H_{m_l, r'_l} \subseteq M, \phi_{m_l}), (H_{m_l, r_l} \subseteq M, \phi_{m_l} \vert_{H_{m_l, r_l}})) \vert l \in L, ((H_{m_l, r'_l} \subseteq M, \phi_{m_l}), (H_{m_l, r_l} \subseteq M, \phi_{m_l} \vert_{H_{m_l, r_l}})) \in \{\text{ the } r' \text{ - } r \text{ -open-half-balls charts pairs around } m_l\}\} (\cup_{j \in J} B_{m_j, r_j} \cup \cup_{l \in L} H_{m_l, r_l} = M)) \text{ where } J, L \in \{\text{ the countable index sets }\}\)
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\(r_j\) and \(r'_j\) can be taken to be any same \(r\) and \(r'\) or they can be taken to be arbitrarily different.
Also the set of the corresponding outer charts constitutes a countable cover of \(M\).
2: Proof
Whole Strategy: Step 1: for each interior point, \(m \in M\), take an \(r'\)-\(r\)-open-balls charts pair around \(m\), \(((B_{m, r'} \subseteq M, \phi_m), (B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}}))\); Step 2: for each boundary point, \(m \in M\), take an \(r'\)-\(r\)-open-half-balls charts pair around \(m\), \(((H_{m, r'} \subseteq M, \phi_m), (H_{m, r} \subseteq M, \phi_m \vert_{H_{m, r}}))\); Step 3: see that \(\{B_{m, r}\} \cup \{H_{m, r}\}\) covers \(M\); Step 4: apply the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover to have a countable cover.
Step 1:
For each interior point, \(m \in M\), let us take any \(r'\)-\(r\)-open-balls charts pair around \(m\), \(((B_{m, r'} \subseteq M, \phi_m), (B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}}))\), which is possible, by the proposition that for any \(C^\infty\) manifold with boundary, each interior point has an \(r'\)-\(r\)-open-balls charts pair and each boundary point has an \(r'\)-\(r\)-open-half-balls charts pair for any positive \(r'\) and \(r\).
Step 2:
For each boundary point, \(m \in M\), let us take any \(r'\)-\(r\)-open-half-balls charts pair around \(m\), \(((H_{m, r'} \subseteq M, \phi_m), (H_{m, r} \subseteq M, \phi_m \vert_{H_{m, r}}))\), which is possible, by the proposition that for any \(C^\infty\) manifold with boundary, each interior point has an \(r'\)-\(r\)-open-balls charts pair and each boundary point has an \(r'\)-\(r\)-open-half-balls charts pair for any positive \(r'\) and \(r\).
Step 3:
\(\{B_{m, r}\} \cup \{H_{m, r}\}\) covers \(M\).
Step 4:
By the proposition that on any 2nd-countable topological space, any open cover of any subset has a countable subcover, there is a countable subcover, \(\{B_{m_j, r_j} \vert j \in J\} \cup \{H_{m_l, r_l} \vert l \in L\}\).
So, \(\{((B_{m_j, r'_j} \subseteq M, \phi_{m_j}), (B_{m_j, r_j} \subseteq M, \phi_{m_j} \vert_{B_{m_j, r_j}})) \vert j \in J\}\) and \(\{((H_{m_l, r'_l} \subseteq M, \phi_{m_l}), (H_{m_l, r_l} \subseteq M, \phi_{m_l} \vert_{H_{m_l, r_l}})) \vert l \in L\}\) satisfy the proposition.
