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.
compactness topological-groups
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up vote
1
down vote
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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.
compactness topological-groups
Are you asking whether all subgroups of a compact group are totally bounded?
â 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.
â Theo Bendit
Sep 3 at 11:27
1
@TheoBendit That makes sense. Thank you.
â 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.
â Daniel Fischerâ¦
Sep 3 at 12:42
@DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
â Sumit Mittal
Sep 3 at 18:47
 |Â
show 2 more comments
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.
compactness topological-groups
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
compactness topological-groups
edited Sep 3 at 11:25
Javi
2,2081725
2,2081725
asked Sep 3 at 10:54
Sumit Mittal
15312
15312
Are you asking whether all subgroups of a compact group are totally bounded?
â 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.
â Theo Bendit
Sep 3 at 11:27
1
@TheoBendit That makes sense. Thank you.
â 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.
â Daniel Fischerâ¦
Sep 3 at 12:42
@DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
â Sumit Mittal
Sep 3 at 18:47
 |Â
show 2 more comments
Are you asking whether all subgroups of a compact group are totally bounded?
â 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.
â Theo Bendit
Sep 3 at 11:27
1
@TheoBendit That makes sense. Thank you.
â 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.
â Daniel Fischerâ¦
Sep 3 at 12:42
@DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
â Sumit Mittal
Sep 3 at 18:47
Are you asking whether all subgroups of a compact group are totally bounded?
â José Carlos Santos
Sep 3 at 11:23
Are you asking whether all subgroups of a compact group are totally bounded?
â 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.
â 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.
â Theo Bendit
Sep 3 at 11:27
1
1
@TheoBendit That makes sense. Thank you.
â José Carlos Santos
Sep 3 at 11:27
@TheoBendit That makes sense. Thank you.
â 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.
â Daniel Fischerâ¦
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.
â Daniel Fischerâ¦
Sep 3 at 12:42
@DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
â Sumit Mittal
Sep 3 at 18:47
@DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
â Sumit Mittal
Sep 3 at 18:47
 |Â
show 2 more comments
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Are you asking whether all subgroups of a compact group are totally bounded?
â 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.
â Theo Bendit
Sep 3 at 11:27
1
@TheoBendit That makes sense. Thank you.
â 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.
â Daniel Fischerâ¦
Sep 3 at 12:42
@DanielFischer as the notion of totally boundedness differs here, compact groups need not be totally bounded groups.
â Sumit Mittal
Sep 3 at 18:47