Subgroups of Compact groups are exactly the totally bounded groups.

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If $G$ is a compact topological group, then how can we prove that subgroups of $G$ are exactly the totally bounded groups.



$textbfNote$: We call a topological group $G$, totally bounded or precompact if $G$ can be covered by finitely many translates of any neighbourhood of identity.










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  • Are you asking whether all subgroups of a compact group are totally bounded?
    – José Carlos Santos
    Sep 3 at 11:23










  • @JoséCarlosSantos As I interpret the question, he wants to show that all of the subgroups of all compact groups are totally bounded, and that every totally bounded group is a subgroup of some compact group. Basically, a group is totally bounded if and only if it is a subgroup of a compact group.
    – Theo Bendit
    Sep 3 at 11:27







  • 1




    @TheoBendit That makes sense. Thank you.
    – José Carlos Santos
    Sep 3 at 11:27










  • If Theo Bendit's interpretation is correct: Every subspace of a totally bounded space is totally bounded, so subgroups of compact groups are totally bounded. The completion of a totally bounded space is compact, and since the completion of a topological group is a topological group, the completion of a totally bounded group is a compact group.
    – Daniel Fischer♦
    Sep 3 at 12:42










  • @DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
    – Sumit Mittal
    Sep 3 at 18:47














up vote
1
down vote

favorite












If $G$ is a compact topological group, then how can we prove that subgroups of $G$ are exactly the totally bounded groups.



$textbfNote$: We call a topological group $G$, totally bounded or precompact if $G$ can be covered by finitely many translates of any neighbourhood of identity.










share|cite|improve this question























  • Are you asking whether all subgroups of a compact group are totally bounded?
    – José Carlos Santos
    Sep 3 at 11:23










  • @JoséCarlosSantos As I interpret the question, he wants to show that all of the subgroups of all compact groups are totally bounded, and that every totally bounded group is a subgroup of some compact group. Basically, a group is totally bounded if and only if it is a subgroup of a compact group.
    – Theo Bendit
    Sep 3 at 11:27







  • 1




    @TheoBendit That makes sense. Thank you.
    – José Carlos Santos
    Sep 3 at 11:27










  • If Theo Bendit's interpretation is correct: Every subspace of a totally bounded space is totally bounded, so subgroups of compact groups are totally bounded. The completion of a totally bounded space is compact, and since the completion of a topological group is a topological group, the completion of a totally bounded group is a compact group.
    – Daniel Fischer♦
    Sep 3 at 12:42










  • @DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
    – Sumit Mittal
    Sep 3 at 18:47












up vote
1
down vote

favorite









up vote
1
down vote

favorite











If $G$ is a compact topological group, then how can we prove that subgroups of $G$ are exactly the totally bounded groups.



$textbfNote$: We call a topological group $G$, totally bounded or precompact if $G$ can be covered by finitely many translates of any neighbourhood of identity.










share|cite|improve this question















If $G$ is a compact topological group, then how can we prove that subgroups of $G$ are exactly the totally bounded groups.



$textbfNote$: We call a topological group $G$, totally bounded or precompact if $G$ can be covered by finitely many translates of any neighbourhood of identity.







compactness topological-groups






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share|cite|improve this question













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edited Sep 3 at 11:25









Javi

2,2081725




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asked Sep 3 at 10:54









Sumit Mittal

15312




15312











  • Are you asking whether all subgroups of a compact group are totally bounded?
    – José Carlos Santos
    Sep 3 at 11:23










  • @JoséCarlosSantos As I interpret the question, he wants to show that all of the subgroups of all compact groups are totally bounded, and that every totally bounded group is a subgroup of some compact group. Basically, a group is totally bounded if and only if it is a subgroup of a compact group.
    – Theo Bendit
    Sep 3 at 11:27







  • 1




    @TheoBendit That makes sense. Thank you.
    – José Carlos Santos
    Sep 3 at 11:27










  • If Theo Bendit's interpretation is correct: Every subspace of a totally bounded space is totally bounded, so subgroups of compact groups are totally bounded. The completion of a totally bounded space is compact, and since the completion of a topological group is a topological group, the completion of a totally bounded group is a compact group.
    – Daniel Fischer♦
    Sep 3 at 12:42










  • @DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
    – Sumit Mittal
    Sep 3 at 18:47
















  • Are you asking whether all subgroups of a compact group are totally bounded?
    – José Carlos Santos
    Sep 3 at 11:23










  • @JoséCarlosSantos As I interpret the question, he wants to show that all of the subgroups of all compact groups are totally bounded, and that every totally bounded group is a subgroup of some compact group. Basically, a group is totally bounded if and only if it is a subgroup of a compact group.
    – Theo Bendit
    Sep 3 at 11:27







  • 1




    @TheoBendit That makes sense. Thank you.
    – José Carlos Santos
    Sep 3 at 11:27










  • If Theo Bendit's interpretation is correct: Every subspace of a totally bounded space is totally bounded, so subgroups of compact groups are totally bounded. The completion of a totally bounded space is compact, and since the completion of a topological group is a topological group, the completion of a totally bounded group is a compact group.
    – Daniel Fischer♦
    Sep 3 at 12:42










  • @DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
    – Sumit Mittal
    Sep 3 at 18:47















Are you asking whether all subgroups of a compact group are totally bounded?
– José Carlos Santos
Sep 3 at 11:23




Are you asking whether all subgroups of a compact group are totally bounded?
– José Carlos Santos
Sep 3 at 11:23












@JoséCarlosSantos As I interpret the question, he wants to show that all of the subgroups of all compact groups are totally bounded, and that every totally bounded group is a subgroup of some compact group. Basically, a group is totally bounded if and only if it is a subgroup of a compact group.
– Theo Bendit
Sep 3 at 11:27





@JoséCarlosSantos As I interpret the question, he wants to show that all of the subgroups of all compact groups are totally bounded, and that every totally bounded group is a subgroup of some compact group. Basically, a group is totally bounded if and only if it is a subgroup of a compact group.
– Theo Bendit
Sep 3 at 11:27





1




1




@TheoBendit That makes sense. Thank you.
– José Carlos Santos
Sep 3 at 11:27




@TheoBendit That makes sense. Thank you.
– José Carlos Santos
Sep 3 at 11:27












If Theo Bendit's interpretation is correct: Every subspace of a totally bounded space is totally bounded, so subgroups of compact groups are totally bounded. The completion of a totally bounded space is compact, and since the completion of a topological group is a topological group, the completion of a totally bounded group is a compact group.
– Daniel Fischer♦
Sep 3 at 12:42




If Theo Bendit's interpretation is correct: Every subspace of a totally bounded space is totally bounded, so subgroups of compact groups are totally bounded. The completion of a totally bounded space is compact, and since the completion of a topological group is a topological group, the completion of a totally bounded group is a compact group.
– Daniel Fischer♦
Sep 3 at 12:42












@DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
– Sumit Mittal
Sep 3 at 18:47




@DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
– Sumit Mittal
Sep 3 at 18:47















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