description/proof of that for group with topology with continuous operations (especially, topological group) and closed normal subgroup, quotient group is topological group
Topics
About: group
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of quotient group of group by normal subgroup.
- The reader knows a definition of closed subset of topological space.
- The reader knows a definition of topological group.
- The reader admits the proposition that for any group with topology with continuous operations (especially, topological group) and any closed subgroup, the left or right cosets of the subgroup quotient topological space is Hausdorff and the classification map is open.
- The reader admits the proposition that for any group, any normal subgroup, and the quotient group by the subgroup, the classification map is a group homomorphism whose kernel the normal subgroup is.
- The reader admits the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
- The reader admits the proposition that the product of any finite number of open quotient maps is open quotient.
Target Context
- The reader will have a description and a proof of the proposition that for any group with topology with continuous operations (especially, any topological group) and any closed normal subgroup, the quotient group with the quotient topology is a topological group.
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:
\(G'\): \(\in \{\text{ the groups with topologies with continuous operations }\}\)
\(G\): \(\in \{\text{ the closed normal subgroups of } G'\}\)
\(G' / G\): \(= \text{ the quotient group with the quotient topology }\)
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Statements:
\(G' / G \in \{\text{ the topological groups }\}\)
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2: Note
Typically, \(G'\) is a topological group, but does not need to be so for this proposition: \(G'\) does not need to be Hausdorff, as it is not used in Proof.
3: Proof
Whole Strategy: Step 1: see that \(G' / G\) is a group and a Hausdorff topological space; Step 2: see that the multiplication map of \(G' / G\) is continuous; Step 3: see that the inversion map of \(G' / G\) is continuous; Step 4: conclude the proposition.
Step 1:
\(G' / G\) is indeed a group, as is seen in Note for the definition of quotient group of group by normal subgroup.
\(G' / G\) with the quotient topology is a Hausdorff topological space and the classification map, \(f: G' \to G' / G\), is an open quotient map, by the proposition that for any group with topology with continuous operations (especially, topological group) and any closed subgroup, the left or right cosets of the subgroup quotient topological space is Hausdorff and the classification map is open.
Step 2:
Let \(m': G' \times G' \to G'\) is the multiplication map.
Let \(\widetilde{m}: G' / G \times G' / G \to G' / G\) is the multiplication map.
\(f \circ m' = \widetilde{m} \circ (f \times f)\), by the proposition that for any group, any normal subgroup, and the quotient group by the subgroup, the classification map is a group homomorphism whose kernel the normal subgroup is.
\(f \times f\) is quotient, by the proposition that the product of any finite number of open quotient maps is open quotient.
\(f \circ m'\) is continuous, by the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
So, \(\widetilde{m} \circ (f \times f)\) is continuous.
Then, \(\widetilde{m}\) is continuous, by the universal property of quotient map: any surjection between topological spaces is a quotient map if and only if any additional map from the codomain of the original map to any additional topological space is continuous if and only if the composition of the additional map after the original map is continuous.
Step 3:
Let \(i': G' \to G'\) be the inversion map.
Let \(\widetilde{i}: G' / G \to G' / G\) be the inversion map.
\(f \circ i' = \widetilde{i} \circ f\), by the proposition that for any group, any normal subgroup, and the quotient group by the subgroup, the classification map is a group homomorphism whose kernel the normal subgroup is.
\(f \circ i'\) is continuous, by the proposition that for any maps between any arbitrary subspaces of any topological spaces continuous at any corresponding points, the composition is continuous at the point.
So, \(\widetilde{i} \circ f\) is continuous.
Then, \(\widetilde{i}\) is continuous, by the universal property of quotient map: any surjection between topological spaces is a quotient map if and only if any additional map from the codomain of the original map to any additional topological space is continuous if and only if the composition of the additional map after the original map is continuous.
Step 4:
So, \(G' / G\) is a topological group.
