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What does 'closed group' mean?
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It's a basic question, but what do you mean when you say that a group or subgroup is closed? Is this that the action of the group over the corresponding space has always a norm less or equal than some number?
If you could give some examples too, it would be great.
group-theory lie-groups
asked yesterday
Vicky
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It's a basic question, but what do you mean when you say that a group or subgroup is closed? Is this that the action of the group over the corresponding space has always a norm less or equal than some number?
If you could give some examples too, it would be great.
group-theory lie-groups
asked yesterday
Vicky
1387
3
It depends on context. A topological (sub)group can be closed if the underlying set is closed in the relevant topology. Or it could just be that the "closure" axiom holds, meaning that the product of two elements from the (sub)group is in the (sub)group.
– T. Bongers
yesterday
I realized my comment was probably a bit hasty, so I edited.
– T. Bongers
yesterday
It is in your best interest that you provide context for your question.
– José Carlos Santos
yesterday
2
In the context of Lie groups, it usually means the underying manifold is closed, i.e. compact and without boundary.
– Pedro Tamaroff♦
yesterday
1
The context you give seems to be another of your own questions — what is the underlying context of your questions?
– Santana Afton
yesterday
|
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up vote
0
down vote
favorite
up vote
0
down vote
favorite
It's a basic question, but what do you mean when you say that a group or subgroup is closed? Is this that the action of the group over the corresponding space has always a norm less or equal than some number?
If you could give some examples too, it would be great.
group-theory lie-groups
asked yesterday
Vicky
1387
It's a basic question, but what do you mean when you say that a group or subgroup is closed? Is this that the action of the group over the corresponding space has always a norm less or equal than some number?
If you could give some examples too, it would be great.
group-theory lie-groups
group-theory lie-groups
asked yesterday
Vicky
1387
asked yesterday
Vicky
1387
asked yesterday
Vicky
1387
asked yesterday
Vicky
1387
asked yesterday
Vicky
1387
1387
3
It depends on context. A topological (sub)group can be closed if the underlying set is closed in the relevant topology. Or it could just be that the "closure" axiom holds, meaning that the product of two elements from the (sub)group is in the (sub)group.
– T. Bongers
yesterday
I realized my comment was probably a bit hasty, so I edited.
– T. Bongers
yesterday
It is in your best interest that you provide context for your question.
– José Carlos Santos
yesterday
2
In the context of Lie groups, it usually means the underying manifold is closed, i.e. compact and without boundary.
– Pedro Tamaroff♦
yesterday
1
The context you give seems to be another of your own questions — what is the underlying context of your questions?
– Santana Afton
yesterday
|
show 3 more comments
3
It depends on context. A topological (sub)group can be closed if the underlying set is closed in the relevant topology. Or it could just be that the "closure" axiom holds, meaning that the product of two elements from the (sub)group is in the (sub)group.
– T. Bongers
yesterday
I realized my comment was probably a bit hasty, so I edited.
– T. Bongers
yesterday
It is in your best interest that you provide context for your question.
– José Carlos Santos
yesterday
2
In the context of Lie groups, it usually means the underying manifold is closed, i.e. compact and without boundary.
– Pedro Tamaroff♦
yesterday
1
The context you give seems to be another of your own questions — what is the underlying context of your questions?
– Santana Afton
yesterday
3
3
It depends on context. A topological (sub)group can be closed if the underlying set is closed in the relevant topology. Or it could just be that the "closure" axiom holds, meaning that the product of two elements from the (sub)group is in the (sub)group.
– T. Bongers
yesterday
It depends on context. A topological (sub)group can be closed if the underlying set is closed in the relevant topology. Or it could just be that the "closure" axiom holds, meaning that the product of two elements from the (sub)group is in the (sub)group.
– T. Bongers
yesterday
I realized my comment was probably a bit hasty, so I edited.
– T. Bongers
yesterday
I realized my comment was probably a bit hasty, so I edited.
– T. Bongers
yesterday
It is in your best interest that you provide context for your question.
– José Carlos Santos
yesterday
It is in your best interest that you provide context for your question.
– José Carlos Santos
yesterday
2
2
In the context of Lie groups, it usually means the underying manifold is closed, i.e. compact and without boundary.
– Pedro Tamaroff♦
yesterday
In the context of Lie groups, it usually means the underying manifold is closed, i.e. compact and without boundary.
– Pedro Tamaroff♦
yesterday
1
1
The context you give seems to be another of your own questions — what is the underlying context of your questions?
– Santana Afton
yesterday
The context you give seems to be another of your own questions — what is the underlying context of your questions?
– Santana Afton
yesterday
|
show 3 more comments
1 Answer
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Unless you are talking about topology or metric spaces, it usually corresponds to an operator on the group
To be closed under the operator means that if you use the operator on any two elements of the group or subgroup, the result is still contained in the group or subgroup. Note that this closure is required for by the definition of a group.
answered yesterday
Mathaddict
1484
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
-2
down vote
Unless you are talking about topology or metric spaces, it usually corresponds to an operator on the group
To be closed under the operator means that if you use the operator on any two elements of the group or subgroup, the result is still contained in the group or subgroup. Note that this closure is required for by the definition of a group.
answered yesterday
Mathaddict
1484
add a comment |
up vote
-2
down vote
Unless you are talking about topology or metric spaces, it usually corresponds to an operator on the group
To be closed under the operator means that if you use the operator on any two elements of the group or subgroup, the result is still contained in the group or subgroup. Note that this closure is required for by the definition of a group.
answered yesterday
Mathaddict
1484
add a comment |
up vote
-2
down vote
up vote
-2
down vote
Unless you are talking about topology or metric spaces, it usually corresponds to an operator on the group
To be closed under the operator means that if you use the operator on any two elements of the group or subgroup, the result is still contained in the group or subgroup. Note that this closure is required for by the definition of a group.
answered yesterday
Mathaddict
1484
Unless you are talking about topology or metric spaces, it usually corresponds to an operator on the group
To be closed under the operator means that if you use the operator on any two elements of the group or subgroup, the result is still contained in the group or subgroup. Note that this closure is required for by the definition of a group.
answered yesterday
Mathaddict
1484
answered yesterday
Mathaddict
1484
answered yesterday
Mathaddict
1484
answered yesterday
Mathaddict
1484
1484
add a comment |
add a comment |
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It depends on context. A topological (sub)group can be closed if the underlying set is closed in the relevant topology. Or it could just be that the "closure" axiom holds, meaning that the product of two elements from the (sub)group is in the (sub)group.
– T. Bongers
yesterday
I realized my comment was probably a bit hasty, so I edited.
– T. Bongers
yesterday
It is in your best interest that you provide context for your question.
– José Carlos Santos
yesterday
2
In the context of Lie groups, it usually means the underying manifold is closed, i.e. compact and without boundary.
– Pedro Tamaroff♦
yesterday
1
The context you give seems to be another of your own questions — what is the underlying context of your questions?
– Santana Afton
yesterday