2024-07-29

704: For C Manifold with Boundary, Interior Point Has r-Open-Ball Chart and Boundary Point Has r-Open-Half-Ball Chart

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description/proof of that for C manifold with boundary, interior point has r-open-ball chart and boundary point has r-open-half-ball chart

Topics


About: C manifold

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that for any C 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.

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: { the d -dimensional C manifolds with boundary }
m: M
r: {rR|0<r}
//

Statements:
(
m{ the interior points of M}

(Bm,rM,ϕm){ the r -open-ball charts around m on M}
)

(
m{ the boundary points of M}

(Hm,rM,ϕm){ the r -open-half-ball charts around m on M}
)
//


2: Proof


Whole Strategy: Step 1: suppose that m is any interior point and take any chart around m, (UmM,ϕm), such that ϕm(Um)=Rd; Step 2: take the r-radius open ball centered at ϕm(m), Bϕm(m),rRd; Step 3: take the preimage of the open ball under the chart map, Bm,r:=ϕm1(Bϕm(m),r), and see that (Bm,rM,ϕm|Bm,r) is a r-open-ball chart; Step 4: suppose that m is any boundary point and take any chart around m, (UmM,ϕm), such that ϕm(Um)=Hd; Step 5: take the r-radius open half ball centered at ϕm(m), Hϕm(m),rHd; Step 6: take the preimage of the open half ball under the chart map, Hm,r:=ϕm1(Hϕm(m),r), and see that (Hm,rM,ϕm|Hm,r) is a r-open-half-ball chart.

Step 1:

Let us suppose that m is any interior point.

Let us take any chart around m, (UmM,ϕm), such that ϕm(Um)=Rd, which is possible by the proposition that for any C manifold with boundary, any interior point has a chart whose range is the whole Euclidean space and any boundary point has a chart whose range is the whole half Euclidean space.

Step 2:

There is the r-radius open ball centered at ϕm(m), Bϕm(m),rRd, which is of course open on Rd.

Step 3:

Let us take the preimage, Bm,r:=ϕm1(Bϕm(m),r)Um.

Bm,r is an open neighborhood of m on Um, because ϕm is homeomorphic. Bm,r is an open neighborhood of m on M, by the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.

(Bm,rM,ϕm|Bm,r) is a chart, because ϕm|Bm,r:Bm,rBϕm(m),r is obviously a homeomorphism and it is C compatible with larger (UmM,ϕm).

ϕm|Bm,r(Bm,r)=Bϕm(m),r.

So, (Bm,rM,ϕm|Bm,r) is an r-open-ball chart.

Step 4:

Let us suppose that m is any boundary point.

Let us take any chart around m, (UmM,ϕm), such that ϕm(Um)=Hd, which is possible by the proposition that for any C manifold with boundary, any interior point has a chart whose range is the whole Euclidean space and any boundary point has a chart whose range is the whole half Euclidean space. ϕm(p) is on the boundary of Hd.

Step 5:

There is the r-radius open half ball centered at ϕm(m), Hϕm(m),rHd, which is of course open on Hd.

Step 6:

Let us take the preimage, Hm,r:=ϕm1(Hϕm(m),r)Um.

Hm,r is an open neighborhood of m on Um, because ϕm is homeomorphic. Hm,r is an open neighborhood of m on M, by the proposition that for any topological space and any topological subspace that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.

(Hm,rM,ϕm|Hm,r) is a chart, because ϕm|Hm,r:Hm,rHϕm(m),r is obviously a homeomorphism and it is C compatible with larger (UmM,ϕm).

ϕm|Hm,r(Hm,r)=Hϕm(m),r.

So, (Hm,rM,ϕm|Hm,r) is an r-open-half-ball chart.


References


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