The authors have previously presented a completion theory for those approach spaces which have an underlying To topology – these include all quasi-metric spaces. This theory extends the existing completion theory for uniform approach spaces, which in turn generalizes that for metric spaces. This new completion theory, moreover, has an interesting relationship with the completion theory for nearness spaces. The theory allows every quasi-metric space to be completed, and remarkably such completions need not again be quasimetric; this situation contrasts with all other previously introduced completion theories for quasi-metric spaces (e.g. [12, 3, 9]). In this paper we present an example of a non-quasi-metric completion, and we give some conditions which ensure that the completion is again quasi-metric. This investigation leads us to favour one particular form of Cauchy sequence in quasi-metric spaces.
