810: Interior C∞ Manifold of C∞ Manifold with Boundary

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definition of interior $C^{\mathrm{\infty }}$ manifold of $C^{\mathrm{\infty }}$ manifold with boundary

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold with boundary.

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Target Context

• The reader will have a definition of interior $C^{\mathrm{\infty }}$ manifold of $C^{\mathrm{\infty }}$ manifold with boundary.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with (possibly empty) boundary }\}$
$\ast intM$: $=\{\text{ the interior points of }M\}$ 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 $M$.
$\wedge$
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 $M$.

$intM$ is contained in the "topology", because $intM\subseteq M$ is an open subset of $M$, because for each $p\in IntM$, 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 $p^{\prime }\in U_p$ is a point of $intM$, because $U_p$ is also an open neighborhood of $p^{\prime }$, so, $U_p\subseteq intM$, and on the other hand, $intM$ contains only interior points of $M$.

For any open subsets, $\{U_{\alpha }\subset intM|\alpha \in A\}$, where $A$ is any possibly uncountable index set, ${\cup }_{\alpha \in A}{U}_{\alpha }$ is an open subset of $intM$, because ${\cup }_{\alpha \in A}{U}_{\alpha }$ is open on $M$ and contains only interior points of $M$.

For any open subsets, $U_1,...,U_k\subseteq intM$, $U_1\cap ...\cap U_k$ is an open subset of $intM$, 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 $intM$, because around each $p\in intM$, there is a chart contained in the "atlas", because while there is a chart, $(U_p\subseteq M,{\varphi }_p)$, contained in the atlas for $M$, there is an open neighborhood of $p$, $U_p^{\prime }\subseteq intM$, on $intM$, and $(U_p\cap U_p^{\prime }\subseteq M,{\varphi }_p{|}_{U_p\cap U_p^{\prime }})$ is a chart for $M$ and $U_p\cap U_p^{\prime }\subseteq intM$, so, it is a chart in the "atlas".

Any 2 charts in the "atlas" are $C^{\mathrm{\infty }}$ compatible, because they are $C^{\mathrm{\infty }}$ compatible in the atlas for $M$.

Any $C^{\mathrm{\infty }}$ compatible chart has been already added into the "atlas", because if there is a chart $C^{\mathrm{\infty }}$ compatible for the "atlas", the chart is $C^{\mathrm{\infty }}$ compatible for the atlas for $M$, so, the chart has been already inherited from the atlas for $M$.

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References

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