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Selections and approaching points in products

Valentin Gutev (2016)

Commentationes Mathematicae Universitatis Carolinae

The present paper aims to furnish simple proofs of some recent results about selections on product spaces obtained by García-Ferreira, Miyazaki and Nogura. The topic is discussed in the framework of a result of Katětov about complete normality of products. Also, some applications for products with a countably compact factor are demonstrated as well.

Selections and suborderability

Giuliano Artico, Umberto Marconi, Jan Pelant, Luca Rotter, Mikhail Tkachenko (2002)

Fundamenta Mathematicae

We extend van Mill-Wattel's results and show that each countably compact completely regular space with a continuous selection on couples is suborderable. The result extends also to pseudocompact spaces if they are either scattered, first countable, or connected. An infinite pseudocompact topological group with such a continuous selection is homeomorphic to the Cantor set. A zero-selection is a selection on the hyperspace of closed sets which chooses always an isolated point of a set. Extending Fujii-Nogura...

Selections and weak orderability

Michael Hrušák, Iván Martínez-Ruiz (2009)

Fundamenta Mathematicae

We answer a question of van Mill and Wattel by showing that there is a separable locally compact space which admits a continuous weak selection but is not weakly orderable. Furthermore, we show that a separable space which admits a continuous weak selection can be covered by two weakly orderable spaces. Finally, we give a partial answer to a question of Gutev and Nogura by showing that a separable space which admits a continuous weak selection admits a continuous selection for all finite sets.

Selections generating new topologies.

Valentin Gutev, Artur Tomita (2007)

Publicacions Matemàtiques

Every (continuous) selection for the non-empty 2-point subsets of a space X naturally defines an interval-like topology on X. In the present paper, we demonstrate that, for a second-countable zero-dimensional space X, this topology may fail to be first-countable at some (or, even any) point of X. This settles some problems stated in [7].

Selections on Ψ -spaces

Michael Hrušák, Paul J. Szeptycki, Artur Hideyuki Tomita (2001)

Commentationes Mathematicae Universitatis Carolinae

We show that if 𝒜 is an uncountable AD (almost disjoint) family of subsets of ω then the space Ψ ( 𝒜 ) does not admit a continuous selection; moreover, if 𝒜 is maximal then Ψ ( 𝒜 ) does not even admit a continuous selection on pairs, answering thus questions of T. Nogura.

Size levels for arcs

Sam Nadler, T. West (1992)

Fundamenta Mathematicae

We determine the size levels for any function on the hyperspace of an arc as follows. Assume Z is a continuum and consider the following three conditions: 1) Z is a planar AR; 2) cut points of Z have component number two; 3) any true cyclic element of Z contains at most two cut points of Z. Then any size level for an arc satisfies 1)-3) and conversely, if Z satisfies 1)-3), then Z is a diameter level for some arc.

Some examples of true F σ δ sets

Marek Balcerzak, Udayan Darji (2000)

Colloquium Mathematicae

Let K(X) be the hyperspace of a compact metric space endowed with the Hausdorff metric. We give a general theorem showing that certain subsets of K(X) are true F σ δ sets.

Spaces of ω-limit sets of graph maps

Jie-Hua Mai, Song Shao (2007)

Fundamenta Mathematicae

Let (X,f) be a dynamical system. In general the set of all ω-limit sets of f is not closed in the hyperspace of closed subsets of X. In this paper we study the case when X is a graph, and show that the family of ω-limit sets of a graph map is closed with respect to the Hausdorff metric.

Striped structures of stable and unstable sets of expansive homeomorphisms and a theorem of K. Kuratowski on independent sets

Hisao Kato (1993)

Fundamenta Mathematicae

We investigate striped structures of stable and unstable sets of expansive homeomorphisms and continuum-wise expansive homeomorphisms. The following theorem is proved: if f : X → X is an expansive homeomorphism of a compact metric space X with dim X > 0, then the decompositions W S ( x ) | x ∈ X and W ( u ) ( x ) | x ∈ X of X into stable and unstable sets of f respectively are uncountable, and moreover there is σ (= s or u) and ϱ > 0 such that there is a Cantor set C in X with the property that for each x ∈ C, W σ ( x ) contains a nondegenerate...

Symmetric products of the Euclidean spaces and the spheres

Naotsugu Chinen (2015)

Commentationes Mathematicae Universitatis Carolinae

By F n ( X ) , n ≥ 1 , we denote the n -th symmetric product of a metric space ( X , d ) as the space of the non-empty finite subsets of X with at most n elements endowed with the Hausdorff metric d H . In this paper we shall describe that every isometry from the n -th symmetric product F n ( X ) into itself is induced by some isometry from X into itself, where X is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the n -th symmetric product of the Euclidean space up to bi-Lipschitz equivalence and...

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