definition of interior \(C^\infty\) manifold of \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
Target Context
- The reader will have a definition of interior \(C^\infty\) manifold of \(C^\infty\) 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:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with (possibly empty) boundary }\}\)
\(*int M\): \(= \{\text{ the interior points of } M\}\) with the topology and the atlas specified below
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Conditions:
The topology is the subset of the topology of \(M\) such that each element of the subset contains only interior points of \(M\).
\(\land\)
The atlas is the subset of the atlas of \(M\) such that the chart domain of each element of the subset contains only interior points of \(M\).
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2: Note
Let us confirm that the "topology" is indeed a topology.
The empty set is contained in the "topology", because it is open on \(M\) and contains (vacuously) only interior points of \(M\).
\(int M\) is contained in the "topology", because \(int M \subseteq M\) is an open subset of \(M\), because for each \(p \in Int M\), there is an open neighborhood of \(p\), \(U_p \subset M\), that is homeomorphic to an open subset of \(int \mathbb{H}^d\), but each \(p' \in U_p\) is a point of \(int M\), because \(U_p\) is also an open neighborhood of \(p'\), so, \(U_p \subseteq int M\), and on the other hand, \(int M\) contains only interior points of \(M\).
For any open subsets, \(\{U_\alpha \subset int M \vert \alpha \in A\}\), where \(A\) is any possibly uncountable index set, \(\cup_{\alpha \in A} U_\alpha\) is an open subset of \(int M\), because \(\cup_{\alpha \in A} U_\alpha\) is open on \(M\) and contains only interior points of \(M\).
For any open subsets, \(U_1, ..., U_k \subseteq int M\), \(U_1 \cap ... \cap U_k\) is an open subset of \(int M\), because \(U_1 \cap ... \cap U_k\) is open on \(M\) and contains only interior points of \(M\).
Let us confirm that the "atlas" is indeed an atlas.
The "atlas" covers \(int M\), because around each \(p \in int M\), there is a chart contained in the "atlas", because while there is a chart, \((U_p \subseteq M, \phi_p)\), contained in the atlas for \(M\), there is an open neighborhood of \(p\), \(U'_p \subseteq int M\), on \(int M\), and \((U_p \cap U'_p \subseteq M, \phi_p \vert_{U_p \cap U'_p})\) is a chart for \(M\) and \(U_p \cap U'_p \subseteq int M\), so, it is a chart in the "atlas".
Any 2 charts in the "atlas" are \(C^\infty\) compatible, because they are \(C^\infty\) compatible in the atlas for \(M\).
Any \(C^\infty\) compatible chart has been already added into the "atlas", because if there is a chart \(C^\infty\) compatible for the "atlas", the chart is \(C^\infty\) compatible for the atlas for \(M\), so, the chart has been already inherited from the atlas for \(M\).
