Category theoretical view of I-cluster and I-limit points of subsequences

We study the concepts of I-limit and I-cluster points of a sequence, where I is an ideal with the Baire property. We obtain the relationship between I-limit and I-cluster points of a subsequence of a given sequence and the set of its classical limit points in the sense of category theory.

GAMES AND HEREDITARY BAIRENESS IN HYPERSPACES AND SPACES OF PROBABILITY MEASURES

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$ , we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$ . Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$ -filter ${\mathcal{F}}$ and prove that it is equivalent to $K({\mathcal{F}})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_{r}(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$ , which is not completely metrizable with $P_{r}(X)$ hereditarily Baire. As far as we know, this is the first example of this kind.

Semigroups of pathological sets

In this article, we demonstrate how the Vitali and Bernstein constructions together with a simple theory on semigroups and ideals of sets can be used for producing different semigroups of sets without the Baire property and/or non-measurable in the Lebesgue sense.

On orbits without the Baire property

The following question is considered: when an uncountable commutative group of homeomorphisms of a second category topological space contains a subgroup, no orbit of which possesses the Baire property?

Porous subsets in the space of functions having the Baire property

The comparison of some subfamilies of the family of functions on the real line having the Baire property in porosity terms is given. We prove that the family of all quasi-continuous functions is strongly porous set in the family of all cliquish functions and that the family of all cliquish functions is strongly porous set in the family of all functions having the Baire property.We prove also that the family of all Świątkowski functions is lower 2/3-porous set in the family of cliquish functions and the family of functions having the internally Świątkowski property is lower 2/3-porous set in the family of cliquish functions.