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Quasi-uniform entropy h QU (ψ) is defined for a uniformly continuous self-map ψ on a T 0 quasi-uniform space (X , U) . Basic properties are proved about this entropy, and it is shown that the quasi-uniform entropy h QU (ψ , U) is less than or equal to the uniform entropy h U (ψ , U s) of ψ considered as a uniformly continuous self-map of the uniform space (X , U s) , where U s is the uniformity associated with the quasi-uniformity U . Finally, we prove that the completion theorem for quasi-uniform entropy holds in the class of all join-compact T 0 quasi-uniform spaces, that is for join-compact T 0 quasi-uniform spaces the entropy of a uniformly continuous self-map coincides with the entropy of its extension to the bicompletion. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to extend the completion theory of a partial metric space to the context of an asymmetric setup, namely, a partial quasi-metric space. We present two bicompletions of a partial quasi-metric space by appealing to the associated partial metric space on one hand and while on the other hand an associated T 0 -quasi-metric space is utilized. The two bicompletions are not necessarily the same but they coincide once the class of partial quasi-metric spaces considered is restricted to the class of T 0 -quasi-metric spaces. [ABSTRACT FROM AUTHOR]

Under the context of quasi-metric, promoted the concept of interval-valued fuzzy metric space, the main results are as follows :(1) topology induced by quasi-metric is consistent with which induced via a standard interval-valued fuzzy quasi-metric; (2) proved that every quasi-metrizable topological space admits a compatible interval-valued fuzzy quasi-metric. On the contrary, topology generated by interval-valued fuzzy quasi-metric is quasi-metrizable; (3) discussed some properties of interval-valued fuzzy quasi-metric space which is bicompletion. proved that if an interval-valued fuzzy quasi-metric space has bicompletion, then it is unique up to isometry. In addition, we define a fuzzy contraction mapping of interval-valued fuzzy metric space, promote the Banach and Edelstein fixed point theorem to interval fuzzy metric space. [ABSTRACT FROM AUTHOR]

The quasi-uniform entropy h Q U (ψ) is defined for a uniformly continuous self-map ψ of a quasi-metric space (X , q). Basic properties are proved about this entropy, and it is shown that the quasi-uniform entropy h Q U (ψ , q) is less or equal to the Bowen's uniform entropy h U (ψ , q s) of ψ considered as a uniformly continuous self-map of the metric space (X , q s) , where q s is the symmetrised metric of the quasi-metric q. Finally, we prove that the completion theorem for the quasi-uniform entropy holds in the class of all join-compact quasi-metric spaces, that is for join-compact quasi-metric spaces the entropy of a uniformly continuous self-map coincides with the entropy of its extension to the bicompletion. [ABSTRACT FROM AUTHOR]

We introduce a quasi-uniformity U on a BCC-algebra X by a family of ideals of X. If T(U) is the topology induced by U, we study some conditions under which (X, T(U)) becomes a (semi)topological BCC-algebra. Also, we show that bicompletion of the quasi-uniformity U can be considered a T(U*)-topological BCC-algebra which contains X as a sub-dense space. [ABSTRACT FROM AUTHOR]

Abstract: This article gives a brief outline of the life of Sergio Salbany (1941–2005) and of his mathematical career, coupled with his social outreach and political activity in South Africa until 1980, in Zimbabwe 1981–1992, in Botswana 1993–1997, and finally again in South Africa 1997–2005. We include a list of his mathematical publications. Salbany was best known for his research in the area of asymmetric topology, comprising topics such as bitopological spaces, quasi-uniform spaces, ordered spaces and quasi-pseudometric spaces. His publications in this area were essentially the first that explicitly described the functors that served to clarify the structure and aims of the then novel theory. [Copyright &y& Elsevier]

Abstract: In [4] Kemajou et al. constructed the injective hull in the category of -quasi-metric spaces with nonexpansive maps that they called q-hyperconvex hull. In this paper, we study properties of functions pairs of the q-hyperconvex hull of asymmetric normed linear spaces. We show, for instance, that any point in the q-hyperconvex hull of an asymmetric normed linear space is convex. [Copyright &y& Elsevier]

Jointly with E. Kemajou, in previous work we constructed the so-called Isbell-hull of a T0-quasi-metric space. In this article we continue these investigations by presenting a similar construction in the category of T0-ultra-quasi-metric spaces and contracting maps. Comparable studies in the area of ultra-metric spaces have been conducted before by Bayod and Martínez-Maurica. [ABSTRACT FROM AUTHOR]

Abstract: We study a concept of hyperconvexity that is appropriate to the category of -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. [Copyright &y& Elsevier]

Abstract: A *-compactification of a quasi-uniform space is a compact quasi-uniform space that has a -dense subspace quasi-isomorphic to . In this paper we study when the hyperspace with the Hausdorff–Bourbaki quasi-uniformity is *-compactifiable and describe some of its *-compactifications. [Copyright &y& Elsevier]

Abstract: Let denote the usual forgetful functor from the category of quasi-uniform -spaces to that of the topological -spaces. We regard the bicompletion reflector as a (pointed) endofunctor . For any section of T we consider the (pointed) endofunctor . 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 . An important known characterisation is that F is upper K-true iff the canonical embedding is an epimorphism in for every X in . 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 , 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. [Copyright &y& Elsevier]

Abstract: 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. [ABSTRACT FROM AUTHOR]

We develop a general technique to study completeness in a metrically generated construct X, the objects of which can be isomorphically described as sets endowed with a gauge of quasi metrics, saturated in a certain way. Our approach depends on a given functor F on X, describing the transition to some known topological construct A in which there already exists a completion theory described by some reflector R and for which also the associated firm class of morphisms $$ \mathfrak{L} $$(R) is known. This functor will permit to “lift” the notion of completeness from A to X. We describe sufficient conditions on R and F, ensuring that the lifting produces a completion theory RF for the construct X0 of T0 objects and that the associated firm class of morphisms can be derived from $$ \mathfrak{L} $$(R). Application of this technique to concrete examples results in many interesting completion theories and provides us with a far better understanding of several existing ones. [ABSTRACT FROM AUTHOR]

Abstract: We continue our study of the conjugate invariant method for completing an arbitrary -quasi-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 -quasi-metric space is totally bounded, but can be strictly larger than the bicompletion. [Copyright &y& Elsevier]

Abstract: We study conditions under which the Hausdorff quasi-uniformity of a quasi-uniform space on the set of the nonempty subsets of X is bicomplete. Indeed we present an explicit method to construct the bicompletion of the -quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform -spaces for which the Hausdorff quasi-uniformity of their bicompletion on is bicomplete. [Copyright &y& Elsevier]

Given a T 0 paratopological group G and a class C of continuous homomorphisms of paratopological groups, we define the C -semicompletion C [ G) and C -completion C [ G ] of the group G that contain G as a dense subgroup, satisfy the T 0 -separation axiom and have certain universality properties. For special classes C , we present some necessary and sufficient conditions on G in order that the (semi)completions C [ G) and C [ G ] be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group G whose C -semicompletion C [ G) fails to be a T 1 -space, where C is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group G contains an ω -bounded sequentially compact subgroup H such that H is a topological group but its closure in G fails to be a subgroup. [ABSTRACT FROM AUTHOR]

Abstract: We present a conjugate invariant method for completing any -quasi-metric space. The completion is built as an extension of the bicompletion of the original space. For balanced -quasi-metric spaces our completion yields up to isometry the completion due to Doitchinov. The question which uniformly continuous maps between -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. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Several results on compactification of quasi-uniform hyperspaces are obtained. For instance, we prove that if C0(X) denotes the family of all nonempty closed subsets of a quasi-uniform space (X,U) and UH the Bourbaki quasi-uniformity of U, then (C0(X),UH) 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, 2βX is equivalent to the *-compactification of (C0(X),PNH), where PN denotes the Pervin quasi-uniformity of X. [Copyright &y& Elsevier]

A biBanach space is an asymmetric normed linear space ( X,|·|) such that the normed linear space ( X,|·|s) is a Banach space, where | x|s= max {| x|,|- x|} for all x∈ X. We prove that each asymmetric normed linear space ( X,|·|) is isometrically isomorphic to a dense subspace of a biBanach space ( Y,|·|Y). Furthermore the space ( Y,|·|Y) is unique (up to isometric isomorphism). [ABSTRACT FROM AUTHOR]

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. [Copyright &y& Elsevier]

It is proved that the quasi-proximity space induced by the bicompletion of a quasi-uniform T0-space X is a subspace of the quasi-proximity space induced by the Samuel bicompactification of X. The result is then used to establish that the locally finite covering quasi-uniformity defined on the category Top0 of topological T0-spaces and continuous maps is not lower K-true (in the sense of Brümmer). It is also shown that a functorial quasi-uniformity F on Top0 is upper K-true if and only if FX is bicomplete whenever X is sober. [ABSTRACT FROM AUTHOR]

We show that the bicompletion of a weightable quasi-metric space is a weightable quasi-metric space. From this result we deduce that any partial metric space has an (up to isometry) unique partial metric bicompletion. Some other consequences are derived. In particular, applications to two interesting examples of partial metric spaces which appear in Computer Science, as the domain of words and the complexity space, are given. [ABSTRACT FROM AUTHOR]