definition of \(r'\)-\(r\)-open-balls charts pair around point on \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of \(r'\)-\(r\)-open-balls charts pair around point on \(C^\infty\) manifold 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:
\( M\): \(\in \{\text{ the } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\( m\): \(\in M\)
\( r'\): \(\in \{r'' \in \mathbb{R} \vert 0 \lt r''\}\)
\( r\): \(\in \{r'' \in \mathbb{R} \vert 0 \lt r''\}\) such that \(r \lt r'\)
\( (B_{m, r'} \subseteq M, \phi_m)\): \(= \text{ the } r' \text{ -open-ball chart }\)
\( (B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}})\): \(= \text{ the } r \text{ -open-ball chart }\) such that \(B_{m, r} \subset B_{m, r'}\)
\(*((B_{m, r'} \subseteq M, \phi_m), (B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}}))\):
//
Conditions:
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\((B_{m, r'} \subseteq M, \phi_m)\) is called "outer (open-ball) chart".
\((B_{m, r} \subseteq M, \phi_m \vert_{B_{m, r}})\) is called "inner (open-ball) chart".
2: Note
There is no \(r'\)-\(r\)-open-balls charts pair around \(m\) when \(m\) is a boundary point.
There is always an \(r'\)-\(r\)-open-balls charts pair around \(m\) for any positive \(r'\) and \(r\) when \(m\) is an interior point, 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 \(r\).
The reason why an \(r'\)-\(r\)-open-balls charts pair is sometimes useful is that \(\overline{B_{m, r}} \subset B_{m, r'}\) while \(\overline{B_{m, r}}\) is the closure on \(B_{m, r'}\) and on \(M\) and is a compact subspace of \(B_{m, r'}\) and \(M\).
Let us see that fact.
On \(B_{m, r'}\), \(\overline{B_{m, r}} = \phi_m^{-1} (\overline{B_{\phi_m (m), r}})\), by the proposition that for any continuous map between any topological spaces, the closure of the map preimage of any subset of the codomain equals the preimage of the closure of the subset if the map is open especially if the map is surjective and any open subset of the domain is the preimage of an open subset of the codomain.
From \(\overline{B_{\phi_m (m), r}} \subset B_{\phi_m (m), r'}\), \(\phi_m^{-1} (\overline{B_{\phi_m (m), r}}) \subset \phi_m^{-1} (B_{\phi_m (m), r'})\), but the left hand side is \(\overline{B_{m, r}}\) and the right hand side is \(B_{m, r'}\).
As \(\overline{B_{\phi_m (m), r}}\) is compact on \(B_{\phi_m (m), r'}\), \(\overline{B_{m, r}}\) is compact on \(B_{m, r'}\).
\(\overline{B_{m, r}}\) is compact on \(M\), by the proposition that for any topological space, any compact subset of any subspace is compact on the base space.
\(\overline{B_{m, r}}\) is closed on \(M\), by the proposition that each compact subset of any Hausdorff topological space is closed.
If the closure of \(B_{m, r}\) on \(M\) was not \(\overline{B_{m, r}}\) but \(C' \subseteq M\), \(C' \subset \overline{B_{m, r}}\) and so, \(C' \cap B_{m, r'} \subset \overline{B_{m, r}}\), but \(C' \cap B_{m, r'}\) would be a closed subset of \(B_{m, r'}\) and \(B_{m, r} \subseteq C' \cap B_{m, r'}\), and so, \(\overline{B_{m, r}}\) such that \(C' \cap B_{m, r'} \subset \overline{B_{m, r}}\) would not be any closure on \(B_{m, r'}\), a contradiction, so, \(\overline{B_{m, r}}\) is the closure on \(M\).
