definition of left or right cosets of subgroup quotient topological space of group with topology with continuous operations (especially, topological group)
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of left or right coset of subgroup by element of group.
- The reader knows a definition of quotient topology on set with respect to map.
Target Context
- The reader will have a definition of left or right cosets of subgroup quotient topological space of group with topology with continuous operations (especially, 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:
\( G'\): \(\in \{\text{ the groups with topologies with continuous operations }\}\)
\( G\): \(\in \{\text{ the subgroups of } G'\}\)
\( \sim_{G, l}\): \(= \{(g'_1, g'_2) \in G' \times G' \vert \exists g' \in G' (g'_1, g'_2 \in g' G)\}\), \(\in \{\text{ the equivalence relations on } G'\}\)
\( \sim_{G, r}\): \(= \{(g'_1, g'_2) \in G' \times G' \vert \exists g' \in G' (g'_1, g'_2 \in G g')\}\), \(\in \{\text{ the equivalence relations on } G'\}\)
\(*G' / \sim_{G, l}\): \(= \text{ the quotient topological space }\)
\(*G' / \sim_{G, r}\): \(= \text{ the quotient topological space }\)
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Conditions:
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2: Note
Each of \(\sim_{G, l}\) and \(\sim_{G, r}\) is indeed an equivalence relation, by the proposition that for any group and any subgroup, being in any same coset is an equivalence relation, and each of \(G' / \sim_{G, l}\) and \(G' / \sim_{G, r}\) is a quotient topological space as is mentioned in Note for the definition of quotient topology on set with respect to map.
Each of \(G' / \sim_{G, l}\) and \(G' / \sim_{G, r}\) is not any group with respect to the canonical multiplication and inversion unless \(G\) is a normal subgroup, by the proposition that for any group and any subgroup, the quotient set by being in same coset is a group with respect to the canonical multiplication and inversion only if the subgroup is a normal subgroup.
\(G'\) can be a topological group being Hausdorff, but does not need to be so for this definition.
