# Search Results

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## Abstract

^{F}for the construct X

_{0}of

*T*

_{0}objects and that the associated firm class of morphisms can be derived from

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## Abstract

V. Gregori and S. Romaguera [17] obtained an example of a fuzzy metric space (in the sense of A. George and P. Veeramani)
that is not completable, i.e. it is not isometric to a dense subspace of any complete fuzzy metric space; therefore, and contrary
to the classical case, there exist quiet fuzzy quasi-metric spaces that are not bicompletable neither *D*-completable, via (quasi-)isometries. In this paper we show that, nevertheless, it is possible to obtain solutions to the
problem of completion of fuzzy quasi-metric spaces by using quasi-uniform isomorphisms instead of (quasi-)isometries. Such
solutions are deduced from a general method, given here, to obtain extension properties of fuzzy quasi-metric spaces from
the corresponding ones of the classical theory of quasi-uniform and quasi-metric spaces.

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## Abstract

On every approach space *X*, we construct a compatible quasi-uniform gauge structure which turns out to be at the same time the coarsest functorial structure
and the finest compatible totally bounded one. Based on the analogy with the classical Császár-Pervin quasi-uniform space,
we call this the “Császár-Pervin” quasi-uniform gauge space. By means of the bicompletion of this Császár-Pervin quasi-uniform
gauge space of a *T*
_{0} approach space *X*, we succeed in constructing the sobrification of *X*.

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## 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.

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## Abstract

We show that each first countable paratopological vector space *X* has a compatible translation invariant quasi-metric such that the open balls are convex whenever *X* is a pseudoconvex vector space. We introduce the notions of a right-bounded subset and of a right-precompact subset of a
paratopological vector space *X* and prove that *X* is quasi-normable if and only if the origin has a convex and right-bounded neighborhood. Duality in this context is also
discussed. Furthermore, it is shown that the bicompletion of any paratopological vector space (respectively, of any quasi-metric
vector space) admits the structure of a paratopological vector space (respectively, of a quasi-metric vector space). Finally,
paratopological vector spaces of finite dimension are considered.

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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.

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*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.

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## 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.