On completion of fuzzy quasi-metric spaces

Following the ideas of D. Doitchinov, a notion of Cauchy sequence in fuzzy quasi-metric spaces is introduced and used to define a completion for a special class of such spaces.     

Asymmetry and bicompletion of approach spaces

We develop a bicompletion theory for the category Ap₀ of T₀ approach spaces in the sense of Lowen [R. Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford University Press, Oxford, 1997], which extends the completion theory obtained in [R. Lowen, K. Robeys., Completions of products of metric spaces, Quart. J. Math. Oxford 43 (1991) 319-338] for the subcategory of Hausdorff uniform approach spaces. Moreover, we prove it to be firmly epireflective (in the sense of [G.C.L. Brümmer, E. Giuli, A categorical concept of completion of objects, Comment. Math. Univ. Carolin. 33 (1992) 131-147]) with respect to a certain morphism class of dense embeddings.     

On completeness in quasi-metric spaces

A notion of Cauchy sequence in quasi-metric spaces is introduced and used to define a standard completion for a special class of spaces.     

Bicompleting weightable quasi-metric spaces and partial metric spaces

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. The second and third listed authors are partially supported by grants from Spanish Ministry of Science and Technology, BFM 2000-III, and Polytechnical University of Valencia.     

Quasi-uniform completions of partially ordered spaces

In this paper we define partially ordered quasi-uniform spaces (X, , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology of a PO-quasi-uniform space (X, , ≤), the bicompletion of (X, ) is also a PO-quasi-uniform space ( , ⪯) with a partial order ⪯ on that extends ≤ in a natural way.      

Completeness in quasi-metric spaces and Ekeland Variational Principle

In this paper we prove a quasi-metric version of Ekeland Variational Principle and study its connections with the completeness properties of the underlying quasi-metric space. The equivalence with Caristi-Kirk?s fixed point theorem and a proof of Clarke?s fixed point theorem for directional contractions within this framework are also considered.     

Mapping spaces and homology isomorphisms

Let denote the space of pointed continuous maps from a finite cell complex to a space . Let be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on and , will send an -isomorphism in either variable to a map that is monic in homology. Interesting examples arise by letting be -theory, the finite complex be a sphere, and the map in the variable be an exotic unstable Adams map between Moore spaces.

     

Nonlinear proximinal multivalued contractions on quasi-metric spaces

In this paper, taking into account the concept of proximinal set, we obtain some new fixed point results for nonlinear proximinal multivalued contractions on some kind of complete quasi-metric spaces. Also, we present some results for multivalued mappings with closed values by adding a condition on the function ensuring the contraction.     

On separation axioms in fuzzy topological spaces,fuzzy neighborhood spaces,and fuzzy uniform spaces

A certain number of separation axioms for fuzzy topological spaces are provided, all of which are good extensions of the topological (T₀), (T₁), or (T₂). All valid implications between the different axioms are studied and counterexamples are given for the nonvalid ones.     

The Banach fixed point theorem in fuzzy quasi-metric spaces with application to the domain of words

We present a fuzzy quasi-metric version of the Banach contraction principle, which constitutes an extension of the famous Grabiec fixed point theorem. By using this result we show the existence of fixed point for contraction mappings on the domain of words when it is endowed with certain fuzzy quasi-metrics of Baire type. We apply this approach to deduce the existence of solution for some recurrence equations associated to the analysis of Quicksort algorithms and Divide & Conquer algorithms, respectively.     

Completeness of hyperspaces of compact subsets of quasi-metric spaces

We study the hyperspace K ₀(X) of non-empty compact subsets of a Smyth-complete quasi-metric space (X, d). We show that K ₀(X), equipped with the Hausdorff quasi-pseudometric H _d forms a (sequentially) Yoneda-complete space. Moreover, if d is a T ₁ quasi-metric, then the hyperspace is algebraic, and the set of all finite subsets forms a base for it. Finally, we prove that K ₀(X), H _d ) is Smyth-complete if (X, d) is Smyth-complete and all compact subsets of X are d ⁻¹-precompact.     

Applications of utility functions defined on quasi-metric spaces

A quasi-metric space (X,d) is called sup-separable if (X,d^s) is a separable metric space, where d^s(x,y)=max{d(x,y),d(y,x)} for all x,y∈X. We characterize those preferences, defined on a sup-separable quasi-metric space, for which there is a semi-Lipschitz utility function. We deduce from our results that several interesting examples of quasi-metric spaces which appear in different fields of theoretical computer science admit semi-Lipschitz utility functions. We also apply our methods to the study of certain kinds of dynamical systems defined on quasi-metric spaces.     

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Let θ ∈ (0, 1), λ ∈ [0, 1) and p, p ₀, p ₁ ∈ (1,∞] be such that (1 ? θ)/p ₀ + θ/p ₁ = 1/p, and let φ, φ₀, φ₁ be some admissible functions such that φ, φ₀ ^p/p0 and φ₁ ^p/p1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L _φ0 ^p0),λ (X), L _φ1 ^{p1), λ} (X), θ> of two generalized grand Morrey spaces on a quasi-metric measure space X is the generalized grand Morrey space L _φ ^p),λ (X). Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.     

Quicksort algorithm: Application of a fixed point theorem in intuitionistic fuzzy quasi-metric spaces at a domain of words

In this paper, we apply an intuitionistic fuzzy quasi-metric version of a fixed point theorem, to obtain the existence of solution for a recurrence equation associated with the analysis of Quicksort algorithms.     

On hyperspace topologies via distance functionals in quasi-metric spaces

Summary Let (X,d) be a quasi-pseudo-metric space. We investigate hyperspace topologies on P₀(X) defined by distance functionals. In particular, the K-topology is introduced and compared with other hyperspace topologies. Some properties of the Wijsman topology and the K-topology are explored.     

On isomorphisms of continuous function spaces

The following theorem, which strengthens the classical theorem of Stone, is proved: If there is an isomorphism φ ofC(X) ontoC(Y) (X, Y-compact) with ∥ φ ∥ · ∥ φ^?1 ∥ < 2, thenX andY are homeomorphic.