Complete partial metric spaces have partially metrizable computational models

We show that the domain of formal balls of a complete partial metric space (X, p) can be endowed with a complete partial metric that extends p and induces the Scott topology. This result, that generalizes well-known constructions of Edalat and Heckmann [A computational model for metric spaces, Theoret. Comput. Sci. 193 (1998), pp. 53-73] and Heckmann [Approximation of metric spaces by partial metric spaces, Appl. Cat. Struct. 7 (1999), pp. 71-83] for metric spaces and improves a recent result of Romaguera and Valero [A quantitative computational model for complete partial metric spaces via formal balls, Math. Struct. Comput. Sci. 19 (2009), pp. 541-563], motivates a notion of a partially metrizable computational model which allows us to characterize those topological spaces that admit a compatible complete partial metric via this model.The authors acknowledge the support of the Spanish Ministry of Science and Innovation, under grant MTM2009-12872-C02-01.Romaguera Bonilla, S.; Tirado Peláez, P.; Valero Sierra, Ó. (2012). Complete partial metric spaces have partially metrizable computational models. International Journal of Computer Mathematics. 89(3):284-290. https://doi.org/10.1080/00207160.2011.559229S284290893ALI-AKBARI, M., HONARI, B., POURMAHDIAN, M., & REZAII, M. M. (2009). The space of formal balls and models of quasi-metric spaces. Mathematical Structures in Computer Science, 19(2), 337-355. doi:10.1017/s0960129509007439Edalat, A., & Heckmann, R. (1998). A computational model for metric spaces. Theoretical Computer Science, 193(1-2), 53-73. doi:10.1016/s0304-3975(96)00243-5Edalat, A., & Sünderhauf, P. (1999). Computable Banach spaces via domain theory. Theoretical Computer Science, 219(1-2), 169-184. doi:10.1016/s0304-3975(98)00288-6Flagg, B., & Kopperman, R. (1997). Computational Models for Ultrametric Spaces. Electronic Notes in Theoretical Computer Science, 6, 151-159. doi:10.1016/s1571-0661(05)80164-1Heckmann, R. (1999). Applied Categorical Structures, 7(1/2), 71-83. doi:10.1023/a:1008684018933Kopperman, R., Künzi, H.-P. A., & Waszkiewicz, P. (2004). Bounded complete models of topological spaces. Topology and its Applications, 139(1-3), 285-297. doi:10.1016/j.topol.2003.12.001Krötzsch, M. (2006). Generalized ultrametric spaces in quantitative domain theory. Theoretical Computer Science, 368(1-2), 30-49. doi:10.1016/j.tcs.2006.05.037Künzi, H.-P. A. (2001). Nonsymmetric Distances and Their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology. History of Topology, 853-968. doi:10.1007/978-94-017-0470-0_3LAWSON, J. (1997). Spaces of maximal points. Mathematical Structures in Computer Science, 7(5), 543-555. doi:10.1017/s0960129597002363Martin, K. (1998). Domain theoretic models of topological spaces. Electronic Notes in Theoretical Computer Science, 13, 173-181. doi:10.1016/s1571-0661(05)80221-xMatthews, S. G.Partial metric topology. Procedings of the 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci. 728 (1994), pp. 183–197Rodríguez-López, J., Romaguera, S., & Valero, O. (2008). Denotational semantics for programming languages, balanced quasi-metrics and fixed points. International Journal of Computer Mathematics, 85(3-4), 623-630. doi:10.1080/00207160701210653Romaguera, S., & Valero, O. (2009). A quasi-metric computational model from modular functions on monoids. International Journal of Computer Mathematics, 86(10-11), 1668-1677. doi:10.1080/00207160802691652ROMAGUERA, S., & VALERO, O. (2009). A quantitative computational model for complete partial metric spaces via formal balls. Mathematical Structures in Computer Science, 19(3), 541-563. doi:10.1017/s0960129509007671ROMAGUERA, S., & VALERO, O. (2010). Domain theoretic characterisations of quasi-metric completeness in terms of formal balls. Mathematical Structures in Computer Science, 20(3), 453-472. doi:10.1017/s0960129510000010Rutten, J. J. M. M. (1998). Weighted colimits and formal balls in generalized metric spaces. Topology and its Applications, 89(1-2), 179-202. doi:10.1016/s0166-8641(97)00224-1Schellekens, M. P. (2003). A characterization of partial metrizability: domains are quantifiable. Theoretical Computer Science, 305(1-3), 409-432. doi:10.1016/s0304-3975(02)00705-3Smyth, M. B. (2006). The constructive maximal point space and partial metrizability. Annals of Pure and Applied Logic, 137(1-3), 360-379. doi:10.1016/j.apal.2005.05.032Waszkiewicz, P. (2003). Applied Categorical Structures, 11(1), 41-67. doi:10.1023/a:1023012924892WASZKIEWICZ, P. (2006). Partial metrisability of continuous posets. Mathematical Structures in Computer Science, 16(02), 359. doi:10.1017/s096012950600519

Matkowski's type theorems for generalized contractions on (ordered) partial metric spaces

[EN] We obtain extensions of Matkowski's fixed point theorem for generalized contractions of Ciric's type on 0-complete partial metric spaces and on ordered 0-complete partial metric spaces, respectively.The author thanks the support of the Ministry of Science and Innovation of Spain, under grant MTM2009-12872-C02-01Romaguera, S. (2011). Matkowski's type theorems for generalized contractions on (ordered) partial metric spaces. Applied General Topology. 12(2):213-220. https://doi.org/10.4995/agt.2011.1653SWORD21322012

Convergence and Completeness in L_2 (P) with respect to a Partial Metric

Metric spaces can be generalized to be partial metric spaces. Partial metric spaces have a unique concept related to a distance. In usual case, there is no distance from two same points. But, we can obtain the distance from two same points in partial metric spaces. It means that the distance is not absolutely zero. Using the basic concept of partial metric spaces, we find analogy between metric spaces and partial metric spaces. We define a metric d^p formed by a partial metric p, with applying characteristics of metric and partial metric. At the beginning, we implement the metric d^p to determine sequences in L_2 (P). We then ensure the convergence and completeness in L_2 [a,b] can be established in L_2 (P). In this study, we conclude that the convergence and completeness in L_2 [a,b]  can be established in L_2 (P) by constructing a partial metric p_2 induced by a metric d^p

common fixed points in a partially ordered partial metric space

In the first part of this paper, we prove some generalized versions of the result of Matthews in (Matthews, 1994) using different types of conditions in partially ordered partial metric spaces for dominated self-mappings or in partial metric spaces for self-mappings. In the second part, using our results, we deduce a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends the Subrahmanyam characterization of metric completeness

A domain-theoretic approach to fuzzy metric spaces

We introduce a partial order (sic)(M) on the set BX of formal balls of a fuzzy metric space (X, M, Lambda) in the sense of Kramosil and Michalek, and discuss some of its properties. We also characterize when the poset (BX, (sic)(M)) is a continuous domain by means of a new notion of fuzzy metric completeness introduced here. The well-known theorem of Edalat and Heckmann that a metric space is complete if and only if its poset of formal balls is a continuous domain, is deduced from our characterizationSupported by the Ministry of Economy and Competitiveness of Spain, under grant MTM2012-37894-C02-01.Ricarte Moreno, LA.; Romaguera Bonilla, S. (2014). A domain-theoretic approach to fuzzy metric spaces. Topology and its Applications. 163:149-159. doi:10.1016/j.topol.2013.10.014S14915916

On the domain of formal balls of the Sorgenfrey quasi-metric space

[EN] We show that the poset of formal balls of the Sorgenfrey quasi-metric space is an omega-continuous domain, and deduce that it is also a computational model, in the sense of R.C. Flagg and R. Kopperman, for the Sorgenfrey line. Furthermore, we study its structure of quantitative domain in the sense of P. Waszkiewicz. (C) 2016 Elsevier B.V. All rights reserved.Supported by the Ministry of Economy and Competitiveness of Spain, under grant MTM2012-37894-C02-01.Romaguera Bonilla, S.; Schellekens, M.; Tirado Peláez, P.; Valero Sierra, Ó. (2016). On the domain of formal balls of the Sorgenfrey quasi-metric space. Topology and its Applications. 203:177-187. https://doi.org/10.1016/j.topol.2015.12.086S17718720

Topology of Function Spaces

This dissertation is a study of the relationship between a topological space X and varioushigher-order objects that we can associate with X. In particular the focus is on C(X), the setof all continuous real-valued functions on X endowed with the topology of pointwise convergence,the compact-open topology and an admissible topology. The topological propertiesof continuous function universals and zero set universals are also examined. The topologicalproperties studied can be divided into three types (i) compactness type properties, (ii) chainconditions and (iii) sequential type properties.The dissertation begins with some general results on universals describing methods ofconstructing universals. The compactness type properties of universals are investigatedand it is shown that the class of metric spaces can be characterised as those with a zeroset universal parametrised by a sigma-compact space. It is shown that for a space to have aLindelof-Sigma zero set universal the space must have a sigma-disjoint basis.A study of chain conditions in Ck(X) and Cp(X) is undertaken, giving necessary andsufficient conditions on a space X such that Cp(X) has calibre (kappa,lambda,mu), with a similar resultobtained for the Ck(X) case. Extending known results on compact spaces it is shown that if aspace X is omega-bounded and Ck(X) has the countable chain condition then X must be metric.The classic problem of the productivity of the countable chain condition is investigated inthe Ck setting and it is demonstrated that this property is productive if the underlying spaceis zero-dimensional. Sufficient conditions are given for a space to have a continuous functionuniversal parametrised by a separable space, ccc space or space with calibre omega1.An investigation of the sequential separability of function spaces and products is undertaken. The main results include a complete characterisation of those spaces such that Cp(X)is sequentially separable and a characterisation of those spaces such that Cp(X) is stronglysequentially separable

Caristi's type mappings on complete partial metric spaces

[EN] We introduce a new type of Caristi's mapping on partial metric spaces and show that a partial metric space is complete if and only if every Caristi mapping has a fixed point. From this result we deduce a characterization of bicomplete weightable quasi-metric spaces. Several illustrative examples are given.The third author thanks the support of the Spanish Ministry of Science and Innovation, grand MTM2009-12872-C02-01.Acar, Ö.; Altun, I.; Romaguera Bonilla, S. (2013). Caristi's type mappings on complete partial metric spaces. Fixed Point Theory. 14(1):3-10. http://hdl.handle.net/10251/56874S31014

Common fixed points of Ciric-type contractions on partial metric spaces

We obtain a common fixed point theorem of Boyd-Wong type for four mappings satisfying a Ciric-type contraction on a complete partial metric space. Our result generalizes and unifies, among others, the very recent results of L. CIRIC, B. SAMET, H. AYDI and C. VETRO [Common fixed points of generalized contractions on partial metric spaces and an application, Appl. Math. Comput., 218 (2011), 2398-2406], S. ROMAGUERA [Fixed point theorems for generalized contractions on partial metric spaces, Topology Appl., 159 (2012), 194-199], T. ABDELJAWAD, E. KARAPINAR and K. TAS [Existence and uniqueness of a common fixed point on partial metric spaces, Appl. Math. Lett. 24 (2011), 1900-1904], and D. ILIC, V. PAVLOVIC and V. RAKOCEVIC [Some new extensions of Banach's contraction principle to partial metric space, Appl. Math. Lett. 24 (2011), 1326-1330].The third named author is supported by the Ministry of Science and Innovation of Spain, grant MTM2009-12872-C02-01.Abbas, M.; Altun, I.; Romaguera Bonilla, S. (2013). Common fixed points of Ciric-type contractions on partial metric spaces. Publicationes Mathematicae Debrecen. 82:425-438. https://doi.org/10.5486/PMD.2013.5342S4254388

Complexity spaces as quantitative domains of computation

We study domain theoretic properties of complexity spaces. Although the so-called complexity space is not a domain for the usual pointwise order, we show that, however, each pointed complexity space is an ¿-continuous domain for which the complexity quasi-metric induces the Scott topology, and the supremum metric induces the Lawson topology. Hence, each pointed complexity space is both a quantifiable domain in the sense of M. Schellekens and a quantitative domain in the sense of P. Waszkiewicz, via the partial metric induced by the complexity quasi-metric. © 2011 Elsevier B.V.Romaguera Bonilla, S.; Schellekens, M.; Valero Sierra, O. (2011). Complexity spaces as quantitative domains of computation. Topology and its Applications. 158:853-860. doi:10.1016/j.topol.2011.01.005S85386015