702: Chart Ball Around Point on C∞ Manifold with Boundary

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definition of chart ball around point on $C^{\mathrm{\infty }}$ 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: Natural Language Description
• 3: Note

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Starting Context

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

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Target Context

• The reader will have a definition of chart ball around point on $C^{\mathrm{\infty }}$ 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:
$M$: $\in \{\text{ the }d\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$p$: $\in M$
$\ast B_p$: $\in \{\text{ the open neighborhoods of }p\text{ on }M\}$
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Conditions:
$\mathrm{\exists }(B_p\subseteq M,{\varphi }_p)\in \{\text{ the charts on }M\}$
$\wedge$
${\varphi }_p(B_p)\in \{\text{ the open balls around }{\varphi }_p(p)\text{ on }{\mathbb{R}}^d\}$
//

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2: Natural Language Description

For any $d$-dimensional $C^{\mathrm{\infty }}$ manifold with boundary, $M$, and any point, $p\in M$, any open neighborhood of $p$, $B_p\subseteq M$, such that there is a chart, $(B_p\subseteq M,{\varphi }_p)$, and ${\varphi }_p(B_p)\subseteq {\mathbb{R}}^d$ is an open ball around ${\varphi }_p(p)$

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3: Note

There is no chart ball around $p$ when $p$ is a boundary point.

There is always a chart ball around $p$ when $p$ is an interior point, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, each interior point has a chart ball and each boundary point has a chart half ball.

This concept is different from 'open ball around point on metric space': on any metric space, $M$, around each $p\in M$, $B_{p,ϵ}:=\{p^{\prime }\in M|dist(p^{\prime },p)<ϵ\}$ is always an open ball, which is not necessarily homeomorphic to any open ball on ${\mathbb{R}}^d$.

It can be called also "chart open ball around $p$ on $M$", but of course, each chart domain is known to be an open subset.

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References

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