Császár (1963) and Deák (1991) have introduced the notion of half-completeness in quasi-uniform spaces which generalizes the well known notion of bicompleteness. In this paper, for any quasi-uniform space, we construct a half-completion, called standard half-completion. The standard half-completion coincides with the usual uniform completion in the case of uniform spaces. It is also an idempotent operation in the sense that the standard half-completion of a half-complete quasi-uniform space coincides (up to a quasi-isomorphism) with the space itself.
We introduce a theory of completeness (the π-completeness) for quasi-uniform spaces which extends the theories of bicompleteness
and half-completeness and prove that every quasi-uniform space has a π-completion. This theory is based on a new notion of
a Cauchy pair of nets which makes use of couples of nets. We call them cuts of nets and our inspiration is due to the construction of the τ-cut on a quasi-uniform space (cf. , ). This new version of
completeness coincides with bicompletion, half-completion and D-completion in extended subclasses of the class of quasi-uniform spaces.