Salvador Romaguera

A Kirk Type Characterization of Completeness for Partial Metric Spaces

We extend the celebrated result of W. A. Kirk that a metric space is complete if and only if every Caristi self-mapping for has a fixed point, to partial metric spaces.

The Hausdorff fuzzy metric on compact sets

We propose a method for constructing a Hausdorff fuzzy metric on the set of the nonempty compact subsets of a given fuzzy metric space (in the sense of George and Veeramani). We discuss several important properties as completeness, completion and precompactness. Some illustrative examples are given.

Characterizing completable fuzzy metric spaces

Recently we showed the existence of a fuzzy metric space, in the sense of George and Veeramani, which is not completable (Fuzzy Sets and Systems 130 (2002) 399). Here we present an internal characterization of those fuzzy metric spaces that are completable. Some applications are derived and some illustrative examples are given.

Quasi-metric properties of complexity spaces

Abstract The complexity (quasi-metric) space has been introduced as a part of the development of a topological foundation for the complexity analysis of algorithms (Schellekens, 1995). Applications of this theory to the complexity analysis of Divide and Conquer algorithms have been discussed by Schellekens (1995). Here we obtain several quasi-metric properties of the complexity space. The main results obtained are the Smyth-completeness of the complexity space and the compactness of closed complexity spaces which possess a (complexity) lower bound. Finally, some implications of these results in connection to the above mentioned complexity analysis techniques are discussed and the total boundedness of complexity spaces with a lower bound is discussed in the light of Smyths computational interpretation of this property (Smyth, 1991).

On completion of fuzzy metric spaces

Completions of fuzzy metric spaces (in the sense of George and Veeramani) are discussed. A complete fuzzy metric space Y is said to be a fuzzy metric completion of a given fuzzy metric space X if X is isometric to a dense subspace of Y. We present an example of a fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a dense subspace uniformly isomorphic to it.

A quantitative computational model for complete partial metric spaces via formal balls

Given a partial metric space (X, p), we use (BX, ⊑dp) to denote the poset of formal balls of the associated quasi-metric space (X, dp). We obtain characterisations of complete partial metric spaces and sup-separable complete partial metric spaces in terms of domain-theoretic properties of (BX, ⊑dp). In particular, we prove that a partial metric space (X, p) is complete if and only if the poset (BX, ⊑dp) is a domain. Furthermore, for any complete partial metric space (X, p), we construct a Smyth complete quasi-metric q on BX that extends the quasi-metric dp such that both the Scott topology and the partial order ⊑dp are induced by q. This is done using the partial quasi-metric concept recently introduced and discussed by H. P. Kunzi, H. Pajoohesh and M. P. Schellekens (Kunzi et al. 2006). Our approach, which is inspired by methods due to A. Edalat and R. Heckmann (Edalat and Heckmann 1998), generalises to partial metric spaces the constructions given by R. Heckmann (Heckmann 1999) and J. J. M. M. Rutten (Rutten 1998) for metric spaces.

Denotational semantics for programming languages, balanced quasi-metrics and fixed points

Abstract A new mathematical model is introduced for the study of the domain of words. We do it by means of the introduction of a suitable balanced quasi-metric on the set of all words over an alphabet. It will be shown that this construction has better quasi-metric and topological properties than several classical constructions. We also prove a fixed point theorem which allows us to develop an application for the study of probabilistic divide and conquer algorithms.

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.

Domain theoretic characterisations of quasi-metric completeness in terms of formal balls†

We characterise those quasi-metric spaces (X, d) whose poset BX of formal balls satisfies the condition (*) \begin{linenomath}\begin{equation} \text{for every } (x,r),(y,s)\in \mathbf{B}X,\ (x,r)\ll (y,s)\Leftrightarrow d(x,y)X, d) is Smyth-complete if and only if BX is a dcpo satisfying condition (*). We also give characterisations in terms of formal balls for sequentially Yoneda complete quasi-metric spaces and for Yoneda complete T1 quasi-metric spaces. Finally, we discuss several properties of the Heckmann quasi-metric on the formal balls of any quasi-metric space.

The bicompletion of an asymmetric normed linear space

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).