1157: -Half-Slice of Chart Domain w.r.t. Point
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definition of -half-slice of chart domain w.r.t. point
Topics
About:
manifold
The table of contents of this article
Starting Context
Target Context
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The reader will have a definition of -half-slice of chart domain with respect to point.
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:
:
:
: ,
: such that
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Conditions:
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2: Note
inevitably exists as a nonempty subset for any and any (as far as is satisfied, which requires that the chart is chosen to allow such a ), because and .
is like , where whether the 1st component is really or and whether the last component is really or depend on .
Let be the projection, where whether the 1st component is really or and whether the last component is really or depend on .
is a diffeomorphism, because it is bijective and (there are the standard chart for and the standard chart for such that the components function is , because the components function has the extension, ), and the inverse is (there are the standard chart for and the standard chart for such that the components function is , because the components function has the extension, the inverse of ). Especially, is a homeomorphism.
When , is a diffeomorphism: it is like where , which is bijective and (there are the standard chart for and the standard chart for such that the components function is , because the components function has the extension, ), and the inverse is (there are the standard chart for and the standard chart for such that the components function is , because the components function has the extension, the inverse of ). Especially, is a homeomorphism.
When and (we do not need the case, because we need this only for when the chart is a boundary chart, which guaranteed that ), is a diffeomorphism: it is like where , which is bijective and (there are the standard chart for and the standard chart for such that the components function is , because the components function has the extension, ), and the inverse is (there are the standard chart for and the standard chart for such that the components function is , because the components function has the extension, the inverse of ). Especially, is a homeomorphism. The condition, , is necessary, because otherwise, , which would not be homeomorphic to .
Let us suppose that the chart is an interior chart.
is an open subset of .
So, is an open subset of .
So, is an open subset of .
is a homeomorphism, because it is , and and are some homeomorphisms as some restrictions of homeomorphic and : as the codomain of is the subspace of , but as is the subspace of , the codomain is the subspace of , by the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace, while as the domain of is the subspace of , but as is the subspace of , the domain is the subspace of , likewise.
Let us suppose that the chart is a boundary chart.
is an open subset of and .
So, is an open subset of .
So, is an open subset of .
is a homeomorphism, because it is , and and are some homeomorphisms as some restrictions of homeomorphic and : as the codomain of is the subspace of , but as is the subspace of , the codomain is the subspace of , as before, while as the domain of is the subspace of , but as is the subspace of , the domain is the subspace of , likewise.
The purpose of requiring is only that has at least boundary point.
It is not that would be critically bad if it had no boundary point, but it would be useless for our purpose, because we are considering in order to allow a boundary point.
But is that requirement not too restrictive? No, we are considering '-half-slice of chart domain with respect to point' in order to have some boundary points (otherwise, '-slice of chart domain with respect to point' would be enough), and we can just choose as one of them and if , we can just translate to make .
The reason why we have elaborated on those facts is that we are going to construct a chart, , for any that satisfies a certain condition (called "local-slice-or-half-slice condition"), to make an embedded submanifold with boundary of .
References
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