Anna Di Concilio

Uniform continuity on bounded sets and the Attouch-Wets topology

Let CL(X) be the nonempty closed subsets of a metrizable space X. If d is a compatible metric, the metrizable Attouch-Wets topology Taw (d) on CL(X) is the topology of uniform convergence of distance functionals associated with elements of CL(X) on bounded subsets of X. The main result of this paper shows that two compatible metrics d and p determine the same Attouch-Wets topologies if and only if they determine the same bounded sets and the same class of functions that are uniformly continpous on bounded sets.

Quasi-metric spaces and point-free geometry

An approach to point-free geometry based on the notion of a quasi-metric is proposed in which the primitives are the regions and a non-symmetric distance between regions. The intended models are the bounded regular closed subsets of a metric space together with the Hausdorff excess measure.

Proximal Set-Open Topologies on Partial Maps

Let X, Y be T1 topological spaces. A partial map from X to Y is a continuous function f whose domain is a subspace D of X and whose codomain is Y. Let P(X, Y) be the set of partial maps with domains in a fixed class D. In analogy with the global case, we introduce on P(X, Y), whatever be the nature of the domain class D, new function space topologies, the proximal set-open topologies, briefly PSOTs, deriving from general networks on X and proximity on Y by replacing inclusion with strong inclusion. The PSOTs include the already known generalized compact-open topology on partial maps with closed domains. When domains are supposed closed, the network α closed and hereditarily closed and the proximity δ on Y Efremovic, then the PSOT attached to α and δ is uniformizable iff α is a Urysohn family in X.

Uniformities, hyperspaces, and normality

LetX be a completely regular space and 2X the hyperspace ofX. It is shown that the uniform topologies on 2X arising from Nachbin uniformity onX, which is the weak uniformity generated byC(X, ℝ), and from Tukey—Shirota uniformity onX, generated by all countable open normal coverings ofX, agree. They, both, coincide with a Vietoris-type topology on 2X, the countable locally finite topology, iffX is normal.

Hypertopologies on ω_µ-Metric spaces

The ωμ−metric spaces, with ωμ a regular ordinal number, are sets equipped with a distance valued in a totally ordered abelian group having as character ωμ, but satisfying the usual formal properties of a real metric. The ωμ−metric spaces fill a large and attractive class of peculiar uniform spaces, those with a linearly ordered base. In this paper we investigate hypertopologies associated with ωμ−metric spaces, in particular the Hausdorff topology induced by the Bourbaki-Hausdorff uniformity associated with their natural underlying uniformity. We show that two ωμ−metrics on a same topological space X induce on the hyperspace CL(X), the set of all non-empty closed sets of X, the same Hausdorff topology if and only if they are uniformly equivalent. Moreover, we explore, again in the ωμ−metric setting, the relationship between the Kuratowski and Hausdorff convergences on CL(X) and prove that an ωμ−sequence {Aα}α<ωμ which admits A as Kuratowski limit converges to A in the Hausdorff topology if and only if the join of A with all Aα is ωμ−compact.