Editing Cauchy sequence (section)

===In topological groups===

Since the topological vector space definition of Cauchy sequence requires only that there be a continuous "subtraction" operation, it can just as well be stated in the context of a [[topological group]]: A sequence <math>(x_k)</math> in a topological group <math>G</math> is a Cauchy sequence if for every open neighbourhood <math>U</math> of the [[Identity element|identity]] in <math>G</math> there exists some number <math>N</math> such that whenever <math>m,n>N</math> it follows that <math>x_n x_m^{-1} \in U.</math> As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in <math>G.</math>

As in the [[Complete metric space#Completion|construction of the completion of a metric space]], one can furthermore define the binary relation on Cauchy sequences in <math>G</math> that <math>(x_k)</math> and <math>(y_k)</math> are equivalent if for every open [[Neighbourhood (mathematics)|neighbourhood]] <math>U</math> of the identity in <math>G</math> there exists some number <math>N</math> such that whenever <math>m,n>N</math> it follows that <math>x_n y_m^{-1} \in U.</math> This relation is an [[equivalence relation]]: It is reflexive since the sequences are Cauchy sequences. It is symmetric since <math>y_n x_m^{-1} = (x_m y_n^{-1})^{-1} \in U^{-1}</math> which by continuity of the inverse is another open neighbourhood of the identity. It is [[Transitive relation|transitive]] since <math>x_n z_l^{-1} = x_n y_m^{-1} y_m z_l^{-1} \in U' U''</math> where <math>U'</math> and <math>U''</math> are open neighbourhoods of the identity such that <math>U'U'' \subseteq U</math>; such pairs exist by the continuity of the group operation.