Camillo Costantini

Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space

Abstract We characterize the separable complete ultrametric spaces whose Wijsman hyperspace admits a continuous selection; such an investigation is closely connected to a similar result of V. Gutev about the Ball hyperspace. The characterization may be obtained in terms of a suitable property either of the base space ( X , d ) ( condition (#)) or of the Wijsman hyperspace itself (total disconnectedness). We also give a necessary and sufficient condition for the zero-dimensionality of the Wijsman hyperspace of a (separable) ultrametric space, and we provide an example where such a hyperspace turns out to be connected.

On the hyperspace of a non-separable metric space.

We prove the following two results. 1) There exists a non-separable complete metric space whose Wijsman hypertopology is not Čech-complete. 2) There exist a non-separable metrizable space and two compatible metrics on it, such that the collections of the Borel sets generated by the relative Wijsman hypertopologies do not coincide.

On the dissonance of some metrizable spaces

Abstract We prove that for every separable, 0-dimensional metrizable space X without isolated points, such that every compact subset of it is scattered, the cocompact topology on the hyperspace of X does not coincide with the upper Kuratowski topology—that is, X is dissonant . In particular, it follows that the rational line is dissonant, and that there exist dissonant, hereditarily Baire, separable metrizable spaces.

An α4 , not Fréchet product of α4 Fréchet spaces

Abstract We shall construct in ZFC two Frechet–Urysohn α 4 -spaces, the product of which is α 4 , but fails to be Frechet–Urysohn. This answers Noguras question from 1985.

Extensions of functions which preserve the continuity on the original domain

Abstract We say that a pair of topological spaces ( X , Y ) is good if for every A ⫅ X and every continuous f : A → Y there exists f :X→Y which extends f and is continuous at every point of A. We use this notion to characterize several classes of topological spaces, as hereditarily normal spaces, hereditarily collectionwise normal spaces, Q-spaces, and completely metrizable spaces. We also show that if X is metrizable and Y is locally compact then ( X , Y ) is good and we answer a question of Arhangelskiis about weakly C-embedded subspaces. For separable metrizable spaces our classification of good pairs is almost complete, e.g., if X is uncountable Polish then ( X , Y ) is good if and only if Y is Polish as well. We also show that if Y is Polish and X metrizable then f can be chosen to be of Baire class 1.

Sequential separability vs selective sequential separability

A space X is sequentially separable if there is a countable D  X such that every point of X is the limit of a sequence of points from D. We present two examples of a sequentially separable space which is not selectively sequentially separable. One of them is in addition countable and sequential.

Compactness and local compactness in hyperspaces

Abstract We give characterizations for a subspace of a hyperspace, endowed with either the Vietoris or the Wijsman topology, to be compact or relatively compact. Then we characterize—both globally and at the single points—the local compactness of a hyperspace, endowed with either the Vietoris, the Wijsman, or the Hausdorff metric topology. Several examples illustrating the behaviour of local compactness are also included.

On the resolvability of locally connected spaces

We solve a problem of Padmavally about resolvability of locally connected spaces, in the case where the space under consideration is regular.

Lebesgue density and exceptional points: LEBESGUE DENSITY AND EXCEPTIONAL POINTS

Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many

Pairwise disjoint eight‐shaped curves in hybrid planes

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