1121: For C∞ Manifold with Boundary, Interior Point Has r′-r-Open-Balls Charts Pair and Boundary Point Has r′-r-Open-Half-Balls Charts Pair

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description/proof of that for $C^{\mathrm{\infty }}$ manifold with boundary, interior point has $r^{\prime }$-$r$-open-balls charts pair and boundary point has $r^{\prime }$-$r$-open-half-balls charts pair

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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: Proof

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Starting Context

• The reader knows a definition of $r^{\prime }$-$r$-open-balls charts pair around point on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of $r^{\prime }$-$r$-open-half-balls charts pair around point on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, each interior point has an $r$-open-ball chart and each boundary point has an $r$-open-half-ball chart for any positive $r$.

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Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, each interior point has an $r^{\prime }$-$r$-open-balls charts pair and each boundary point has an $r^{\prime }$-$r$-open-half-balls charts pair for any positive $r^{\prime }$ and $r$.

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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:
$M$: $\in \{\text{ the }d\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$m$: $\in M$
$r^{\prime }$: $\in \{r^{″}\in \mathbb{R}|0<r^{″}\}$
$r$: $\in \{r^{″}\in \mathbb{R}|0<r^{″}\}$ such that $r<r^{\prime }$
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Statements:
(
$m\in \{\text{ the interior points of }M\}$
$⟹$
$\mathrm{\exists }((B_{m,r^{\prime }}\subseteq M,{\varphi }_m),(B_{m,r}\subseteq M,{\varphi }_m{|}_{B_{m,r}}))\in \{\text{ the }{r}^{\prime }\text{ - }r\text{ -open-balls charts pairs around }m\text{ on }M\}$
)
$\wedge$
(
$m\in \{\text{ the boundary points of }M\}$
$⟹$
$\mathrm{\exists }((H_{m,r^{\prime }}\subseteq M,{\varphi }_m),(H_{m,r}\subseteq M,{\varphi }_m{|}_{H_{m,r}}))\in \{\text{ the }{r}^{\prime }\text{ - }r\text{ -open-half-balls charts pairs around }m\text{ on }M\}$
)
//

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2: Proof

Whole Strategy: Step 1: suppose that $m$ is any interior point, and take any $r^{\prime }$-open-ball chart around $m$, $(B_{m,r^{\prime }}\subseteq M,{\varphi }_m)$; Step 2: take $B_{{\varphi }_m(m),r}\subset B_{{\varphi }_m(m),r^{\prime }}$ and define $B_{m,r}:={\varphi }_m^{-1}(B_{{\varphi }_m(m),r})$; Step 3: see that $(B_{m,r}\subseteq M,{\varphi }_m{|}_{B_{m,r}})$ is an $r$-open-ball chart around $m$; Step 4: suppose that $m$ is any boundary point, and take any $r^{\prime }$-open-half-ball chart around $m$, $(H_{m,r^{\prime }}\subseteq M,{\varphi }_m)$; Step 5: take $H_{{\varphi }_m(m),r}\subset H_{{\varphi }_m(m),r^{\prime }}$ and define $H_{m,r}:={\varphi }_m^{-1}(H_{{\varphi }_m(m),r})$; Step 6: see that $(H_{m,r}\subseteq M,{\varphi }_m{|}_{H_{m,r}})$ is an $r$-open-half-ball chart around $m$.

Step 1:

Let us suppose that $m$ is any interior point.

Let us take any $r^{\prime }$-open-ball chart around $m$, $(B_{m,r^{\prime }}\subseteq M,{\varphi }_m)$, which is possible, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, each interior point has an $r$-open-ball chart and each boundary point has an $r$-open-half-ball chart for any positive $r$.

Step 2:

Let us take $B_{{\varphi }_m(m),r}\subset B_{{\varphi }_m(m),r^{\prime }}$.

Let us define $B_{m,r}:={\varphi }_m^{-1}(B_{{\varphi }_m(m),r})\subset B_{m,r^{\prime }}$.

$B_{m,r}$ is an open neighborhood of $m$ on $B_{m,r}$ and on $M$.

Step 3:

$(B_{m,r}\subseteq M,{\varphi }_m{|}_{B_{m,r}})$ is a chart, because ${\varphi }_m{|}_{B_{m,r}}:B_{m,r}\to B_{{\varphi }_m(m),r}$ is a homeomorphism and $(B_{m,r}\subseteq M,{\varphi }_m{|}_{B_{m,r}})$ is $C^{\mathrm{\infty }}$ compatible with larger $(B_{m,r^{\prime }}\subseteq M,{\varphi }_m)$.

So, $(B_{m,r}\subseteq M,{\varphi }_m{|}_{B_{m,r}})$ is an $r$-open-ball chart around $m$.

So, $((B_{m,r^{\prime }}\subseteq M,{\varphi }_m),(B_{m,r}\subseteq M,{\varphi }_m{|}_{B_{m,r}}))$ is an $r^{\prime }$-$r$-open-balls charts pair around $m$.

Step 4:

Let us suppose that $m$ is any boundary point.

Let us take any $r^{\prime }$-open-half-ball chart around $m$, $(H_{m,r^{\prime }}\subseteq M,{\varphi }_m)$, which is possible, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary, each interior point has an $r$-open-ball chart and each boundary point has an $r$-open-half-ball chart for any positive $r$.

Step 5:

Let us take $H_{{\varphi }_m(m),r}\subset H_{{\varphi }_m(m),r^{\prime }}$.

Let us define $H_{m,r}:={\varphi }_m^{-1}(H_{{\varphi }_m(m),r})\subset H_{m,r^{\prime }}$.

$H_{m,r}$ is an open neighborhood of $m$ on $H_{m,r}$ and on $M$.

Step 6:

$(H_{m,r}\subseteq M,{\varphi }_m{|}_{H_{m,r}})$ is a chart, because ${\varphi }_m{|}_{H_{m,r}}:H_{m,r}\to H_{{\varphi }_m(m),r}$ is a homeomorphism and $(H_{m,r}\subseteq M,{\varphi }_m{|}_{H_{m,r}})$ is $C^{\mathrm{\infty }}$ compatible with larger $(H_{m,r^{\prime }}\subseteq M,{\varphi }_m)$.

So, $(H_{m,r}\subseteq M,{\varphi }_m{|}_{H_{m,r}})$ is an $r$-open-half-ball chart around $m$.

So, $((H_{m,r^{\prime }}\subseteq M,{\varphi }_m),(H_{m,r}\subseteq M,{\varphi }_m{|}_{H_{m,r}}))$ is an $r^{\prime }$-$r$-open-half-balls charts pair around $m$.

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References

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