Is $hat{G}$ is complete with respect to the induced topology of $G$?
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For a topological group $G$ and a given fundamental system of neighbourhoods of $G$ we can define the completion of G and we call it $hat{G}$. The induced fundamental system of neighbourhoods of $hat{G}$ is given by this Topology induced by the completion of a topological group.
Then Can we say that $hat{G}$ is complete ? i.e., every Cauchy sequence in $hat{G}$ is convergent ?
For this we assume that ${z_n}$ be any Cauchy sequence in $hat{G},$ then given any open neighbourhood $tilde{N}$ of $hat{G}$ there exists an integer $k$ such that whenever $m,n geq k,$ $z_m-z_n in tilde{N}.$ Then how can I show that ${z_n}$ is convergent ? That is we are looking for an element $s in hat{G}$ such that for any neighbourhood $hat{P}$ of $hat{G},$ $z_n in s+hat{P}$ for large $n$. In general will it hold ? I need some help. Thanks.
abstract-algebra general-topology commutative-algebra topological-groups formal-completions
asked 6 hours ago
user371231
541510
add a comment |
up vote
4
down vote
favorite
For a topological group $G$ and a given fundamental system of neighbourhoods of $G$ we can define the completion of G and we call it $hat{G}$. The induced fundamental system of neighbourhoods of $hat{G}$ is given by this Topology induced by the completion of a topological group.
Then Can we say that $hat{G}$ is complete ? i.e., every Cauchy sequence in $hat{G}$ is convergent ?
For this we assume that ${z_n}$ be any Cauchy sequence in $hat{G},$ then given any open neighbourhood $tilde{N}$ of $hat{G}$ there exists an integer $k$ such that whenever $m,n geq k,$ $z_m-z_n in tilde{N}.$ Then how can I show that ${z_n}$ is convergent ? That is we are looking for an element $s in hat{G}$ such that for any neighbourhood $hat{P}$ of $hat{G},$ $z_n in s+hat{P}$ for large $n$. In general will it hold ? I need some help. Thanks.
abstract-algebra general-topology commutative-algebra topological-groups formal-completions
asked 6 hours ago
user371231
541510
Normally you'd have to consider Cauchy nets for full completeness. You seem to only want sequential completeness?
– Henno Brandsma
6 hours ago
Is the completion not the completion of the uniform structure on $G$?
– Robert Thingum
4 hours ago
Yes, but if $G$ is not first-countable then the completion of $G$, as a uniform space, in general is not determined by Cauchy sequences alone. Sequential completeness only suffices if $G$ is first-countable.
– Monstrous Moonshiner
3 hours ago
Right, I didn't mean to imply that sequences were sufficient. Just wanted to clarify what the completion was.
– Robert Thingum
3 hours ago
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
For a topological group $G$ and a given fundamental system of neighbourhoods of $G$ we can define the completion of G and we call it $hat{G}$. The induced fundamental system of neighbourhoods of $hat{G}$ is given by this Topology induced by the completion of a topological group.
Then Can we say that $hat{G}$ is complete ? i.e., every Cauchy sequence in $hat{G}$ is convergent ?
For this we assume that ${z_n}$ be any Cauchy sequence in $hat{G},$ then given any open neighbourhood $tilde{N}$ of $hat{G}$ there exists an integer $k$ such that whenever $m,n geq k,$ $z_m-z_n in tilde{N}.$ Then how can I show that ${z_n}$ is convergent ? That is we are looking for an element $s in hat{G}$ such that for any neighbourhood $hat{P}$ of $hat{G},$ $z_n in s+hat{P}$ for large $n$. In general will it hold ? I need some help. Thanks.
abstract-algebra general-topology commutative-algebra topological-groups formal-completions
asked 6 hours ago
user371231
541510
For a topological group $G$ and a given fundamental system of neighbourhoods of $G$ we can define the completion of G and we call it $hat{G}$. The induced fundamental system of neighbourhoods of $hat{G}$ is given by this Topology induced by the completion of a topological group.
Then Can we say that $hat{G}$ is complete ? i.e., every Cauchy sequence in $hat{G}$ is convergent ?
For this we assume that ${z_n}$ be any Cauchy sequence in $hat{G},$ then given any open neighbourhood $tilde{N}$ of $hat{G}$ there exists an integer $k$ such that whenever $m,n geq k,$ $z_m-z_n in tilde{N}.$ Then how can I show that ${z_n}$ is convergent ? That is we are looking for an element $s in hat{G}$ such that for any neighbourhood $hat{P}$ of $hat{G},$ $z_n in s+hat{P}$ for large $n$. In general will it hold ? I need some help. Thanks.
abstract-algebra general-topology commutative-algebra topological-groups formal-completions
abstract-algebra general-topology commutative-algebra topological-groups formal-completions
asked 6 hours ago
user371231
541510
asked 6 hours ago
user371231
541510
edited 4 hours ago
asked 6 hours ago
user371231
541510
asked 6 hours ago
user371231
541510
asked 6 hours ago
user371231
541510
541510
Normally you'd have to consider Cauchy nets for full completeness. You seem to only want sequential completeness?
– Henno Brandsma
6 hours ago
Is the completion not the completion of the uniform structure on $G$?
– Robert Thingum
4 hours ago
Yes, but if $G$ is not first-countable then the completion of $G$, as a uniform space, in general is not determined by Cauchy sequences alone. Sequential completeness only suffices if $G$ is first-countable.
– Monstrous Moonshiner
3 hours ago
Right, I didn't mean to imply that sequences were sufficient. Just wanted to clarify what the completion was.
– Robert Thingum
3 hours ago
add a comment |
Normally you'd have to consider Cauchy nets for full completeness. You seem to only want sequential completeness?
– Henno Brandsma
6 hours ago
Is the completion not the completion of the uniform structure on $G$?
– Robert Thingum
4 hours ago
Yes, but if $G$ is not first-countable then the completion of $G$, as a uniform space, in general is not determined by Cauchy sequences alone. Sequential completeness only suffices if $G$ is first-countable.
– Monstrous Moonshiner
3 hours ago
Right, I didn't mean to imply that sequences were sufficient. Just wanted to clarify what the completion was.
– Robert Thingum
3 hours ago
Normally you'd have to consider Cauchy nets for full completeness. You seem to only want sequential completeness?
– Henno Brandsma
6 hours ago
Normally you'd have to consider Cauchy nets for full completeness. You seem to only want sequential completeness?
– Henno Brandsma
6 hours ago
Is the completion not the completion of the uniform structure on $G$?
– Robert Thingum
4 hours ago
Is the completion not the completion of the uniform structure on $G$?
– Robert Thingum
4 hours ago
Yes, but if $G$ is not first-countable then the completion of $G$, as a uniform space, in general is not determined by Cauchy sequences alone. Sequential completeness only suffices if $G$ is first-countable.
– Monstrous Moonshiner
3 hours ago
Yes, but if $G$ is not first-countable then the completion of $G$, as a uniform space, in general is not determined by Cauchy sequences alone. Sequential completeness only suffices if $G$ is first-countable.
– Monstrous Moonshiner
3 hours ago
Right, I didn't mean to imply that sequences were sufficient. Just wanted to clarify what the completion was.
– Robert Thingum
3 hours ago
Right, I didn't mean to imply that sequences were sufficient. Just wanted to clarify what the completion was.
– Robert Thingum
3 hours ago
add a comment |
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Normally you'd have to consider Cauchy nets for full completeness. You seem to only want sequential completeness?
– Henno Brandsma
6 hours ago
Is the completion not the completion of the uniform structure on $G$?
– Robert Thingum
4 hours ago
Yes, but if $G$ is not first-countable then the completion of $G$, as a uniform space, in general is not determined by Cauchy sequences alone. Sequential completeness only suffices if $G$ is first-countable.
– Monstrous Moonshiner
3 hours ago
Right, I didn't mean to imply that sequences were sufficient. Just wanted to clarify what the completion was.
– Robert Thingum
3 hours ago