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[EN] Based on previous results that study the completion of fuzzy metric spaces, we show that every intuitionistic fuzzy quasi-metric space, using the notion of fuzzy metric space in the sense of Kramosil and Michalek to obtain a generalization to the quasi-metric setting, has a bicompletion which is unique up to isometry., Pedro Tirado acknowledges the support of the Ministry of Economy and Comptetitiveness of Spain, gran MTM2012-37894-C02-01

We study a concept of hyperconvexity that is appropriate to the category of T 0 -quasi-metric spaces (called di-spaces in the following) and nonexpansive maps. An explicit construction of the corresponding hull (called Isbell-convex hull or, more briefly, Isbell-hull) of a di-space is provided.

Let T : QU 0 → Top 0 denote the usual forgetful functor from the category of quasi-uniform T 0 -spaces to that of the topological T 0 -spaces. We regard the bicompletion reflector as a (pointed) endofunctor K : QU 0 → QU 0 . For any section F : Top 0 → QU 0 of T we consider the (pointed) endofunctor R = T K F : Top 0 → Top 0 . The T -section F is called upper bicompletion-true (briefly, upper K-true ) if the quasi-uniform space KFX is finer than FRX for every X in Top 0 . An important known characterisation is that F is upper K -true iff the canonical embedding X → R X is an epimorphism in Top 0 for every X in Top 0 . We show that this result admits a simple, purely categorical formulation and proof, independent of the setting of quasi-uniform and topological spaces. We thus mention a few other settings where the result is applicable. Returning then to the setting T : QU 0 → Top 0 , we prove: Any T -section F is upper K -true iff for all X the bitopology of KFX equals that of FRX ; and iff the join topology of KFX equals the strong topology (also called the b - or Skula topology) of RX .

Extending the well-known result that every fuzzy metric space, in the sense of Kramosil and Michalek, has a completion which is unique up to isometry, we show that every KM-fuzzy quasi-metric space has a bicompletion which is unique up to isometry, and deduce that for each KM-fuzzy quasi-metric space, the completion of its induced fuzzy metric space coincides with the fuzzy metric space induced by its bicompletion. © 2010 Elsevier B.V., Supported by the Spanish Ministry of Science and Innovation, under Grant MTM2009-12872-C02-01.

In this article we introduce and investigate the concept of a partial quasi-metric and some of its applications. We show that many important constructions studied in Matthews's theory of partial metrics can still be used successfully in this more general setting. In particular, we consider the bicompletion of the quasi-metric space that is associated with a partial quasi-metric space and study its applications in groups and BCK-algebras.

Extensions of CS-regular maps from a dense subset of a metric space to the whole space are discussed in the literature. It is the purpose of this paper to present results on extensions of these maps to the bicompletion in separated quasipseudo metric spaces, and in addition we discuss extensions of maps that preserve some special property. Finally, an analogue of the Tietze extension theorem will be discussed in this context. Keywords: Bicompletion; Cauchy preserving maps; quasi-pseudo metric space; bounded sets; map extensions; quasi-uniform continuity; quasi-uniform continuity on bounded setsQuaestiones Mathematicae 28(2005), 391– 400.

We introduce a theory of completeness (the -completeness) for quasi-uniform spaces which extends the theories of bicompleteness and halfcompleteness and prove that every quasi-uniform space has a -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 -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. Acta Mathematica Hungarica

Several results on compactification of quasi-uniform hyperspaces are obtained. For instance, we prove that if C 0 (X) denotes the family of all nonempty closed subsets of a quasi-uniform space (X, U ) and U H the Bourbaki quasi-uniformity of U , then ( C 0 (X), U H ) is ∗-compactifiable if and only if (X, U ) is closed symmetric and ∗-compactifiable and U −1 is hereditarily precompact. We deduce that for any normal Hausdorff space X , 2 βX is equivalent to the ∗-compactification of ( C 0 (X), PN H ) , where PN denotes the Pervin quasi-uniformity of X .

A ∗-compactification of a T1 quasi-uniform space (X, U ) is a compact T1 quasi-uniform space (Y, V ) that has a T ( V ∗ ) -dense subspace quasi-isomorphic to (X, U ), where V ∗ denotes the coarsest uniformity finer than V . With the help of the notion of T1 ∗-half completion of a quasi-uniform space, which is introduced and studied here, we show that if a T1 quasi-uniform space (X, U ) has a ∗-compactification, then it is unique up to quasi-isomorphism. We identify the ∗-compactification of (X, U ) with the subspace of its bicompletion ( X , U ) consisting of all points which are closed in ( X , T ( U )) and prove that (X, U ) is ∗-compactifiable if and only if it is point symmetric and ( X , U ) is compact. Finally, we discuss some properties of locally fitting T0 quasi-uniform spaces, a large class of quasi-uniform spaces whose bicompletion is T1, and, hence, they are T1 ∗-half completable.

A *-compactification of a T 0 quasi-uniform space ( X , U ) is a compact T 0 quasi-uniform space ( Y , V ) that has a T ( V ∨ V − 1 ) -dense subspace quasi-isomorphic to ( X , U ) . In this paper we study when the hyperspace with the Hausdorff–Bourbaki quasi-uniformity is *-compactifiable and describe some of its *-compactifications.

This is a survey for the working topologist of several categorical aspects of the bicompletion of functorial quasiuniformities. We consider functors $F\::\:\mathbf{Top_{o}}\longrightarrow\mathbf{QU_{o}}$ from the $T_{o}$-topological spaces to the $T_{o}$-quasi-uniform spaces which endow the $T_{o}$-spaces with compatible quasi-unformities. Regarding the bicompletion as a functor $K:\mathbf{QU_{o}}\longrightarrow\mathbf{QU_{o}}$, we ask when the composite R=TKF is an epireflection in $\mathbf{Top_{o}}$ and when the equality KF=FR holds. Thereby we obtain analogues of important classical results from the theory of uniform spaces. We also present some new results concerning weaker versions of the above questions, e.g. when the pointed endofunctor given by TKF can be augmented to a monad. We prove that every epireflective subcategory of $\mathbf{Top_{o}}$ between the subcategory of sober spaces and the subcategory of topologically becomplete spaces can be obtained from a reflexion of the type TKF. We give full proofs of all new results and of some less known result whose proofs in the literature are in some way inaccessible. The exposition is intended for readers with little knowledge of category theory.

For aT3.5-ordered space, certain families of maps are designated as “defining families.“ For each such defining family we construct the smallestT2-ordered compactification such that each member of the family can be extended to the compactification space. Each defining family also generates a quasi-uniformity on the space whose bicompletion produces the sameT2-ordered compactification.

We present a conjugate invariant method for completing any T0-quasi-metric space. The completion is built as an extension of the bicompletion of the original space. For balanced T0-quasi-metric spaces our completion yields up to isometry the completion due to Doitchinov. The question which uniformly continuous maps between T0-quasi-metric spaces can be extended to the constructed completions leads us to introduce and investigate a new class of maps, which we call balanced maps.

We continue our study of the conjugate invariant method for completing an arbitrary T 0quasi-metric space which we have introduced in an earlier article under the name of the B-completion. We present examples that show that B-completeness of quasi-pseudometric spaces is not preserved under quasi-uniform isomorphisms. This leads to investigations of how some well-known operations applied to quasi-pseudometrics affect balancedness of their Cauchy filter pairs. We also observe that the B-completion of a totally bounded T 0quasi-metric space is totally bounded, but can be strictly larger than the bicompletion. © 2009 Elsevier B.V. All rights reserved.

We define a functor from topological quasi-uniform spaces and continuous, uniformly continuous maps to quasi-uniform spaces and uniformly continuous maps. This functor retains the same underlying set, and generalizes a construction due to Junnila (1978) and Kunzi and Ferrario (1991) to obtain another quasi-uniformity, with respect to which the continuous uniformly continuous functions on the original structure are uniformly continuous; the functor then takes the maps to themselves. As a result, this functor takes the Smyth completion of a topological quasi-uniform space into the bicompletion of its image under the functor, and a topological quasi-uniform space is Smyth complete exactly when its image is bicomplete.