 As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in formula_51.As in the construction of the completion of a metric space, one can furthermore define the binary relation on Cauchy sequences in formula_51 that formula_50 and formula_60 are equivalent if for every open neighbourhood formula_52 of the identity in formula_51 there exists some number formula_11 such that whenever formula_55 it follows that formula_65.This relation is an equivalence relation