# Search Results

## You are looking at 1 - 4 of 4 items for

- Author or Editor: Salvador Romaguera x

- Refine by Access: All Content x

## 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.

*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

The authors study quasi-uniformities that are generated by a family of weightable quasi-pseudometrics. Each totally bounded quasi-uniformity is of this kind. In some sense, which is described in this article, a weightable quasi-uniformity is fairly symmetric, with the associated weights generating small symmetrizers.

In the second part of the article we continue our investigations by generalizing a subclass of weightable quasi-uniformities to a more abstract level. We introduce the concept of a *t*-symmetrizable quasi-uniformity, that is, a quasi-uniformity possessing the property that there exists a totally bounded quasi-uniformity such that is a uniformity. It turns out that *t*-symmetrizable quasi-uniformities are closely related to quasi-uniformities generated by weightable quasi-pseudometrics possessing bounded weight functions. We show that several results that were originally proved for weightable quasi-pseudometrics (with bounded weights) still hold in a such apparently broader context.