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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...

Set-theoretic constructions of two-point sets

Ben Chad, Robin Knight, Rolf Suabedissen (2009)

Fundamenta Mathematicae

A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice....

Seven characterizations of non-meager 𝖯-filters

Kenneth Kunen, Andrea Medini, Lyubomyr Zdomskyy (2015)

Fundamenta Mathematicae

We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager -filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager -filter. Here, we identify a filter with a subspace of 2 ω through characteristic functions. Along the way, we generalize to non-meager -filters a result of Miller (1984) about -points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem...

Some examples of continuous images of Radon-Nikodým compact spaces

Alexander D. Arvanitakis, Antonio Avilés (2009)

Czechoslovak Mathematical Journal

We provide a characterization of continuous images of Radon-Nikodým compacta lying in a product of real lines and model on it a method for constructing natural examples of such continuous images.

Some examples related to colorings

Michael van Hartskamp, Jan van Mill (2000)

Commentationes Mathematicae Universitatis Carolinae

We complement the literature by proving that for a fixed-point free map f : X → X the statements (1) f admits a finite functionally closed cover 𝒜 with f [ A ] ∩ A = ∅ for all A ∈ 𝒜 (i.e., a coloring) and (2) β f is fixed-point free are equivalent. When functionally closed is weakened to closed, we show that normality is sufficient to prove equivalence, and give an example to show it cannot be omitted. We also show that a theorem due to van Mill is sharp: for every n ≥ 2 we construct a strongly zero-dimensional Tychonov space...

Some remarks on Radon-Nikodym compact spaces

Alexander D. Arvanitakis (2002)

Fundamenta Mathematicae

The class of quasi Radon-Nikodým compact spaces is introduced. We prove that this class is closed under countable products and continuous images. It includes the Radon-Nikodým compact spaces. Adapting Alster's proof we show that every quasi Radon-Nikodým and Corson compact space is Eberlein. This generalizes earlier results by J. Orihuela, W. Schachermayer, M. Valdivia and C. Stegall. Further the class of almost totally disconnected spaces is defined and it is shown that every quasi Radon-Nikodým...

Spaces not distinguishing convergences

Miroslav Repický (2000)

Commentationes Mathematicae Universitatis Carolinae

In the present paper we introduce a convergence condition ( Σ ' ) and continue the study of “not distinguish” for various kinds of convergence of sequences of real functions on a topological space started in [2] and [3]. We compute cardinal invariants associated with introduced properties of spaces.

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