# Search Results

## Abstract

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. [1], [20]). This new version of
completeness coincides with bicompletion, half-completion and *D-*completion in extended subclasses of the class of quasi-uniform spaces.

*T*

_{0}quasi-uniform space (

*X, U*) we mean a compact

*T*

_{0}quasi-uniform space (

*Y, V*) that has a

*T*(

*V*∨

*V*

^{−1})-dense subspace quasi-isomorphic to (

*X, U*). We prove that (

*X, U*) has a *-compactification if and only if its

*T*

_{0}biocompletion

*X, U*) and (

*X*∪

*G*(

*X*),

_{ X ∪ G ( X ) }) is its minimal *-compactification, where

*G*(

*X*) is the set of all points of

*T*(

*T*

_{2}compactifications of Tychonoff spaces). Applications of our results to some examples in theoretical computer science are given.

## Abstract

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 *T*
_{0}-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.