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.
By a *-compactification of a
quasi-uniform space (
) we mean a compact
quasi-uniform space (
) that has a
)-dense subspace quasi-isomorphic to (
). We prove that (
) has a *-compactification if and only if its
)-closed (we remark that as partial order of *-compactifications we use the inverse of the partial order used for
compactifications of Tychonoff spaces). Applications of our results to some examples in theoretical computer science are given.
We investigate the left-sided scale and the two-sided scale of a quasi-uniform space. While the two-sided scale of a quasi-uniform
space X shows a behavior similar to the usual hyperspace of X equipped with its Hausdorff quasiuniformity, the left-sided scale generalizes the quasi-uniform multifunction space of X into itself.
Either construction of the scale relies on the concept of the prefilter space of a quasi-uniform space. Prefilter spaces of
quasi-uniform spaces are proved to be bicomplete. Consequently both the left-sided and the two-sided scale of a quasiuniform
space are bicomplete. Indeed these scales can be used to construct the bicompletion of the T0-refiection of the Hausdorff quasi-uniformity of a quasiuniform space.
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.