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What is a complete topological group?
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Given a metric space $ X $, one can forms its completion $ hatX $. However, I have seen that one can also define completeness for topological groups. Can someone explain to me (as simply as possible) what complete topological groups are?
I would also appreciate a more conceptual explanation that is not the most straightforward.
topological-groups
asked Aug 1 at 22:57
kgs
1636
add a comment |Â
up vote
1
down vote
favorite
Given a metric space $ X $, one can forms its completion $ hatX $. However, I have seen that one can also define completeness for topological groups. Can someone explain to me (as simply as possible) what complete topological groups are?
I would also appreciate a more conceptual explanation that is not the most straightforward.
topological-groups
asked Aug 1 at 22:57
kgs
1636
They're just topological groups which, when looked at as just a topological space (= forget the group structure), are complete in the usual sense.
– Noah Schweber
Aug 1 at 23:56
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Given a metric space $ X $, one can forms its completion $ hatX $. However, I have seen that one can also define completeness for topological groups. Can someone explain to me (as simply as possible) what complete topological groups are?
I would also appreciate a more conceptual explanation that is not the most straightforward.
topological-groups
asked Aug 1 at 22:57
kgs
1636
Given a metric space $ X $, one can forms its completion $ hatX $. However, I have seen that one can also define completeness for topological groups. Can someone explain to me (as simply as possible) what complete topological groups are?
I would also appreciate a more conceptual explanation that is not the most straightforward.
topological-groups
asked Aug 1 at 22:57
kgs
1636
asked Aug 1 at 22:57
kgs
1636
asked Aug 1 at 22:57
kgs
1636
asked Aug 1 at 22:57
kgs
1636
1636
They're just topological groups which, when looked at as just a topological space (= forget the group structure), are complete in the usual sense.
– Noah Schweber
Aug 1 at 23:56
add a comment |Â
They're just topological groups which, when looked at as just a topological space (= forget the group structure), are complete in the usual sense.
– Noah Schweber
Aug 1 at 23:56
They're just topological groups which, when looked at as just a topological space (= forget the group structure), are complete in the usual sense.
– Noah Schweber
Aug 1 at 23:56
They're just topological groups which, when looked at as just a topological space (= forget the group structure), are complete in the usual sense.
– Noah Schweber
Aug 1 at 23:56
add a comment |Â
1 Answer
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up vote
3
down vote
accepted
A Cauchy sequence in a topological group $G$ is a sequence $a_n$ such that for any neighborhood $U$ of the identity $ein G$, there exists some $Nin mathbbN$ such that $a_na_m^-1in U$ for all $n,mgeq N$. This generalizes the idea of a Cauchy sequence in a metric space, in the sense that if I go far enough out in the sequence, all terms will be "close" to each other, where here the idea of closeness of two elements is characterized by their difference lying inside a neighborhood of the identity. Then a complete topological group is one in which all Cauchy sequences converge to a point in the space (as usual). If the topology on the group is metrizable, this should coincide with the usual idea of completeness in a metric space.
For more reading, see:
https://en.wikipedia.org/wiki/Cauchy_sequence#Generalizations
https://en.wikipedia.org/wiki/Complete_metric_space#Alternatives_and_generalizations
https://en.wikipedia.org/wiki/Uniform_space
answered Aug 1 at 23:35
Taylor M
1564
Good answer. I like looking at the situation from the uniform-space angle.
– Lubin
Aug 2 at 0:32
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
A Cauchy sequence in a topological group $G$ is a sequence $a_n$ such that for any neighborhood $U$ of the identity $ein G$, there exists some $Nin mathbbN$ such that $a_na_m^-1in U$ for all $n,mgeq N$. This generalizes the idea of a Cauchy sequence in a metric space, in the sense that if I go far enough out in the sequence, all terms will be "close" to each other, where here the idea of closeness of two elements is characterized by their difference lying inside a neighborhood of the identity. Then a complete topological group is one in which all Cauchy sequences converge to a point in the space (as usual). If the topology on the group is metrizable, this should coincide with the usual idea of completeness in a metric space.
For more reading, see:
https://en.wikipedia.org/wiki/Cauchy_sequence#Generalizations
https://en.wikipedia.org/wiki/Complete_metric_space#Alternatives_and_generalizations
https://en.wikipedia.org/wiki/Uniform_space
answered Aug 1 at 23:35
Taylor M
1564
Good answer. I like looking at the situation from the uniform-space angle.
– Lubin
Aug 2 at 0:32
add a comment |Â
up vote
3
down vote
accepted
A Cauchy sequence in a topological group $G$ is a sequence $a_n$ such that for any neighborhood $U$ of the identity $ein G$, there exists some $Nin mathbbN$ such that $a_na_m^-1in U$ for all $n,mgeq N$. This generalizes the idea of a Cauchy sequence in a metric space, in the sense that if I go far enough out in the sequence, all terms will be "close" to each other, where here the idea of closeness of two elements is characterized by their difference lying inside a neighborhood of the identity. Then a complete topological group is one in which all Cauchy sequences converge to a point in the space (as usual). If the topology on the group is metrizable, this should coincide with the usual idea of completeness in a metric space.
For more reading, see:
https://en.wikipedia.org/wiki/Cauchy_sequence#Generalizations
https://en.wikipedia.org/wiki/Complete_metric_space#Alternatives_and_generalizations
https://en.wikipedia.org/wiki/Uniform_space
answered Aug 1 at 23:35
Taylor M
1564
Good answer. I like looking at the situation from the uniform-space angle.
– Lubin
Aug 2 at 0:32
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
A Cauchy sequence in a topological group $G$ is a sequence $a_n$ such that for any neighborhood $U$ of the identity $ein G$, there exists some $Nin mathbbN$ such that $a_na_m^-1in U$ for all $n,mgeq N$. This generalizes the idea of a Cauchy sequence in a metric space, in the sense that if I go far enough out in the sequence, all terms will be "close" to each other, where here the idea of closeness of two elements is characterized by their difference lying inside a neighborhood of the identity. Then a complete topological group is one in which all Cauchy sequences converge to a point in the space (as usual). If the topology on the group is metrizable, this should coincide with the usual idea of completeness in a metric space.
For more reading, see:
https://en.wikipedia.org/wiki/Cauchy_sequence#Generalizations
https://en.wikipedia.org/wiki/Complete_metric_space#Alternatives_and_generalizations
https://en.wikipedia.org/wiki/Uniform_space
answered Aug 1 at 23:35
Taylor M
1564
A Cauchy sequence in a topological group $G$ is a sequence $a_n$ such that for any neighborhood $U$ of the identity $ein G$, there exists some $Nin mathbbN$ such that $a_na_m^-1in U$ for all $n,mgeq N$. This generalizes the idea of a Cauchy sequence in a metric space, in the sense that if I go far enough out in the sequence, all terms will be "close" to each other, where here the idea of closeness of two elements is characterized by their difference lying inside a neighborhood of the identity. Then a complete topological group is one in which all Cauchy sequences converge to a point in the space (as usual). If the topology on the group is metrizable, this should coincide with the usual idea of completeness in a metric space.
For more reading, see:
https://en.wikipedia.org/wiki/Cauchy_sequence#Generalizations
https://en.wikipedia.org/wiki/Complete_metric_space#Alternatives_and_generalizations
https://en.wikipedia.org/wiki/Uniform_space
answered Aug 1 at 23:35
Taylor M
1564
answered Aug 1 at 23:35
Taylor M
1564
answered Aug 1 at 23:35
Taylor M
1564
answered Aug 1 at 23:35
Taylor M
1564
1564
Good answer. I like looking at the situation from the uniform-space angle.
– Lubin
Aug 2 at 0:32
add a comment |Â
Good answer. I like looking at the situation from the uniform-space angle.
– Lubin
Aug 2 at 0:32
Good answer. I like looking at the situation from the uniform-space angle.
– Lubin
Aug 2 at 0:32
Good answer. I like looking at the situation from the uniform-space angle.
– Lubin
Aug 2 at 0:32
add a comment |Â
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They're just topological groups which, when looked at as just a topological space (= forget the group structure), are complete in the usual sense.
– Noah Schweber
Aug 1 at 23:56