1198: For Topological Group, Neighborhoods Basis at Satisfies These Properties and Point Multiplied by Neighborhoods Basis at Is Neighborhoods Basis at Point
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description/proof of that for topological group, neighborhoods basis at satisfies these properties and point multiplied by neighborhoods basis at is neighborhoods basis at point
Topics
About:
topological group
The table of contents of this article
Starting Context
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The reader knows a definition of topological group.
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The reader knows a definition of neighborhoods basis at point on topological space.
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The reader knows a definition of inverse of subset of group.
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The reader admits the proposition that for any topological space and its any point, the intersection of any finite number of neighborhoods of the point is a neighborhood of the point.
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The reader admits the proposition that for any group with any topology with any continuous operations (especially, topological group), any element, and any neighborhood of the element, there is a symmetric neighborhood of such that the element multiplied from left by the neighborhood of and multiplied from right by the inverse of the neighborhood of is contained in the neighborhood of the element.
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The reader admits the proposition that for any group, the inverse of any subset is the image of the subset under the inverse map, and the double inverse of the subset is the subset.
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The reader admits the proposition that for any group with any topology with any continuous operations (especially, topological group) and each element, the inversion map, the multiplication-by-element-from-left-or-right map, and the conjugation-by-element map are homeomorphisms.
Target Context
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The reader will have a description and a proof of the proposition that for any topological group, any neighborhoods basis at satisfies these properties and each point multiplied by the neighborhoods basis at is a neighborhoods basis at the 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:
:
:
//
Statements:
1)
2)
3)
4)
5)
//
means ; means .
2: Proof
Whole Strategy: use the Hausdorff-ness and the continuousness of the operations; Step 1: see that 1) holds; Step 2: see that 2) holds; Step 3: see that 3) holds; Step 4: see that 4) holds; Step 5: see that 5) holds; Step 6: see that is a neighborhoods basis at ; Step 7: see that is a neighborhoods basis at .
Step 1:
Let us see that 1) holds.
As is Hausdorff, there is an open neighborhood of , , and an open neighborhood of , , such that .
There is a such that , by the definition of neighborhoods basis at point on topological space.
, so, .
Step 2:
Let us see that 2) holds.
is a neighborhood of , by the proposition that for any topological space and its any point, the intersection of any finite number of neighborhoods of the point is a neighborhood of the point.
So, there is a such that , by the definition of neighborhoods basis at point on topological space.
Step 3:
Let us see that 3) holds.
There is a (symmetric) neighborhood of , , such that , by the proposition that for any group with any topology with any continuous operations (especially, topological group), any element, and any neighborhood of the element, there is a symmetric neighborhood of such that the element multiplied from left by the neighborhood of and multiplied from right by the inverse of the neighborhood of is contained in the neighborhood of the element: in the proposition is taken to be and because is symmetric.
But .
There is an such that , by the definition of neighborhoods basis at point on topological space.
, by the proposition that for any group, the inverse of any subset is the image of the subset under the inverse map, and the double inverse of the subset is the subset.
So, .
So, .
Step 4:
Let us see that 4) holds.
The conjugation map by , , is a homeomorphism, by the proposition that for any group with any topology with any continuous operations (especially, topological group) and each element, the inversion map, the multiplication-by-element-from-left-or-right map, and the conjugation-by-element map are homeomorphisms.
, because and .
So, is a neighborhood of : while contains an open neighborhood of , , contains , which is an open neighborhood of : .
So, there is an such that , by the definition of neighborhoods basis at point on topological space.
Step 5:
Let us see that 5) holds.
The multiplication-by-element-from-right map by , , is a homeomorphism, by the proposition that for any group with any topology with any continuous operations (especially, topological group) and each element, the inversion map, the multiplication-by-element-from-left-or-right map, and the conjugation-by-element map are homeomorphisms.
, because and .
So, is a neighborhood of : contains , which is an open neighborhood of .
So, there is an such that , by the definition of neighborhoods basis at point on topological space.
So, .
Step 6:
Let be any.
Let be any neighborhood of .
is a neighborhood of , because while the multiplication-by--from-left-map is a homeomorphism by the proposition that for any group with any topology with any continuous operations (especially, topological group) and each element, the inversion map, the multiplication-by-element-from-left-or-right map, and the conjugation-by-element map are homeomorphisms, an open neighborhood of contained in is mapped into as an open neighborhood of .
So, there is an such that .
So, , while is a neighborhood of , because while the multiplication-by--from-left-map is a homeomorphism by the proposition that for any group with any topology with any continuous operations (especially, topological group) and each element, the inversion map, the multiplication-by-element-from-left-or-right map, and the conjugation-by-element map are homeomorphisms, an open neighborhood of contained in is mapped into as an open neighborhood of .
Step 7:
Let be any.
Let be any neighborhood of .
is a neighborhood of , because while the multiplication-by--from-right-map is a homeomorphism by the proposition that for any group with any topology with any continuous operations (especially, topological group) and each element, the inversion map, the multiplication-by-element-from-left-or-right map, and the conjugation-by-element map are homeomorphisms, an open neighborhood of contained in is mapped into as an open neighborhood of .
So, there is an such that .
So, , while is a neighborhood of , because while the multiplication-by--from-right-map is a homeomorphism by the proposition that for any group with any topology with any continuous operations (especially, topological group) and each element, the inversion map, the multiplication-by-element-from-left-or-right map, and the conjugation-by-element map are homeomorphisms, an open neighborhood of contained in is mapped into as an open neighborhood of .
References
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