704: For Manifold with Boundary, Interior Point Has -Open-Ball Chart and Boundary Point Has -Open-Half-Ball Chart
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description/proof of that for manifold with boundary, interior point has -open-ball chart and boundary point has -open-half-ball chart
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About:
manifold
The table of contents of this article
Starting Context
Target Context
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The reader will have a description and a proof of the proposition that for any manifold with boundary, each interior point has an -open-ball chart and each boundary point has an -open-half-ball chart for any positive .
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.
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2: Proof
Whole Strategy: Step 1: suppose that is any interior point and take any chart around , , such that ; Step 2: take the -radius open ball centered at , ; Step 3: take the preimage of the open ball under the chart map, , and see that is a -open-ball chart; Step 4: suppose that is any boundary point and take any chart around , , such that ; Step 5: take the -radius open half ball centered at , ; Step 6: take the preimage of the open half ball under the chart map, , and see that is a -open-half-ball chart.
Step 1:
Let us suppose that is any interior point.
Let us take any chart around , , such that , which is possible by the proposition that for any 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 -radius open ball centered at , , which is of course open on .
Step 3:
Let us take the preimage, .
is an open neighborhood of on , because is homeomorphic. is an open neighborhood of on , 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.
is a chart, because is obviously a homeomorphism and it is compatible with larger .
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So, is an -open-ball chart.
Step 4:
Let us suppose that is any boundary point.
Let us take any chart around , , such that , which is possible by the proposition that for any 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. is on the boundary of .
Step 5:
There is the -radius open half ball centered at , , which is of course open on .
Step 6:
Let us take the preimage, .
is an open neighborhood of on , because is homeomorphic. is an open neighborhood of on , 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.
is a chart, because is obviously a homeomorphism and it is compatible with larger .
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So, is an -open-half-ball chart.
References
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