Andrzej Kucharski

On functors preserving skeletal maps and skeletally generated compacta

A map f:X→Y between topological spaces is skeletal if the preimage f−1(A) of each nowhere dense subset A⊂Y is nowhere dense in X. We prove that a normal functor F:Comp→Comp is skeletal (which means that F preserves skeletal epimorphisms) if and only if for any open surjective map f:X→Y between metrizable zero-dimensional compacta with two-element non-degeneracy set Nf={x∈X:|f−1(f(x))|>1} the map Ff:FX→FY is skeletal. This characterization implies that each open normal functor is skeletal. The converse is not true even for normal functors of finite degree. The other main result of the paper says that each normal functor F:Comp→Comp preserves the class of skeletally generated compacta. This contrasts with the known Scepinʼs result saying that a normal functor is open if and only if it preserves the class of openly generated compacta.

Universally Kuratowski–Ulam Spaces And Open-Open Games

Abstract We examine the class of spaces in which the second player has a winning strategy in the open-open game. We show that this spaces are not universally Kuratowski–Ulam. We also show that the games G and G7 introduced by P. Daniels, K. Kunen, H. Zhou [Fund. Math. 145 (1994), no. 3, 205–220] are not equivalent.

A spectral characterization of skeletal maps

We prove that a map between two realcompact spaces is skeletal if and only if it is homeomorphic to the limit map of a skeletal morphism between ω-spectra with surjective limit projections.

Skeletally Dugundji spaces

We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. Our main result states that the following conditions are equivalent for a given space X: (i) X is skeletally Dugundji; (ii) every compactification of X is co-absolute to a Dugundji space; (iii) every C*-embedding of the absolute p(X) in another space is strongly π-regular; (iv) X has a multiplicative lattice in the sense of Shchepin [Shchepin E.V., Topology of limit spaces with uncountable inverse spectra, Uspekhi Mat. Nauk, 1976, 31(5), 191–226 (in Russian)] consisting of skeletal maps.

On Open-Open Games of Uncountable Length

The aim of this paper is to investigate the open-open game of uncountable length. We introduce a cardinal number 𝜇(𝑋), which says how long the Player I has to play to ensure a victory. It is proved that 𝑐(𝑋)≤𝜇(𝑋)≤𝑐(𝑋)

Game theoretic approach to skeletally Dugundji and Dugundji spaces

Abstract Characterizations of skeletally Dugundji spaces and Dugundji spaces are given in terms of club collections, consisting of countable families of co-zero sets. For example, a Tychonoff space X is skeletally Dugundji if and only if there exists an additive c-club on X. Dugundji spaces are characterized by the existence of additive d-clubs.

Parovičenko spaces with structures

We study an analogue of the Parovičenko property in categories of compact spaces with additional structures. In particular, we present an internal characterization of this property in the class of compact median spaces.

Boolean algebras admitting a countable minimally acting group

The aim of this paper is to show that every infinite Boolean algebra which admits a countable minimally acting group contains a dense projective subalgebra.

Extension of Vladimirov's lemma

Abstract In the paper we consider an extension of Vladimirovs lemma on independent sets in complete Boolean algebras. As an application we obtain a criterion for existence of maximal independent families in compact spaces.