T_0*-compactification in the hyperspace

A *-compactification of a T0 quasi-uniform space (X,U) is a compact T0 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.Kunzi, HA.; Romaguera Bonilla, S.; Sanchez Granero, MA. (2012). T_0*-compactification in the hyperspace. Topology and its Applications. 159:1815-1819. doi:10.1016/j.topol.2011.06.064S1815181915

Quasi-uniform hyperspaces of compact subsets

AbstractLet (X,u) be a quasi-uniform space, K(X) be the family of all nonempty compact subsets of (X,u). In this paper, the notion of compact symmetry for (X,u) is introduced, and relationships between the Bourbaki quasi-uniformity and the Vietoris topology on K(X) are examined. Furthermore we establish that for a compactly symmetric quasi-uniform space (X,u) the Bourbaki quasi-uniformity u∗ on K(X) is complete if and only if u is complete. This theorem generalizes the well-known Zenor-Morita theorem for uniformisable spaces to the quasi-uniform setting

Hyperspaces of a weightable quasi-metric space: Application to models in the theory of computation

It is well known that both weightable quasi-metrics and the Hausdorff distance provide efficient tools in several areas of Computer Science. This fact suggests, in a natural way, the problem of when the upper and lower Hausdorff quasi-pseudo-metrics of a weightable quasi-metric space (X,d) are weightable. Here we discuss this problem. Although the answer is negative in general, we show, however, that it is positive for several nice classes of (nonempty) subsets of X. Since the construction of these classes depends, to a large degree, on the specialization order of the quasi-metric d, we are able to apply our results to some distinguished quasi-metric models that appear in theoretical computer science and information theory, like the domain of words, the interval domain and the complexity space.The authors thank the referees for their useful remarks and comments. The first author was supported by the South African Research Foundation under grant FA2006022300009. The second and third author was supported by the Spanish Ministry of Science and Innovation, under grant MTM2009-12872-C02-01 (subprogram MTM).Künzi, H.; Rodríguez López, J.; Romaguera Bonilla, S. (2010). Hyperspaces of a weightable quasi-metric space: Application to models in the theory of computation. Mathematical and Computer Modelling. 52:674-682. https://doi.org/10.1016/j.mcm.2010.04.015S6746825

On realcompact topological vector spaces

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