on the topology of the topological completion

Asked Jun 27 '21 at 14:40

Active Jun 27 '21 at 16:31

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Let be $G$ a topological group and take $\bar{G}$ its completion, namely the set of cauchy sequence quotiented by the natural equivalence relation. The completion is a group and so if I want that it is a topological group is sufficent to show open neighborhoods of the origin (I see this group additively written). I would want to know if is right to describe the neighborhood of the origin like the set of the kind

$\{$class of Cauchy sequence such that a representant is ultimately in U$\}$

where $U$ is a generic open neighborhood of the origin in $G$

edited Jun 27 '21 at 16:31

Eric Wofsey

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asked Jun 27 '21 at 14:40

LuckyS

What is a Cauchy sequence on a topological group? – José Carlos Santos Jun 27 '21 at 14:42
It's a numerable sequence of elements such that given an open neighborhood of 0 exists an index such that... – LuckyS Jun 27 '21 at 14:43
Got it. Thanks. – José Carlos Santos Jun 27 '21 at 14:44
For a general topological group, sequences are not enough. You will have to use Cauchy nets or Cauchy filters instead. If $G$ is metrizable, though, sequences will be enough. Another complication: if $G$ is not Abelian, can we talk about left-Cauchy and right-Cauchy sequences? – GEdgar Jun 27 '21 at 17:23
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Previous question on completion of an abelian topological group: https://math.stackexchange.com/q/192808/442 – GEdgar Jun 27 '21 at 17:25

on the topology of the topological completion

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