Papers by Lyubomyr Zdomsky
Archive for Mathematical Logic
We prove that CH implies the existence of a Cohen-indestructible mad family such that the Mathias... more We prove that CH implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while b = c is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel's conjecture. Theorem 1.1. p = cov (N) = c implies the existence of a Cohen-indestructible mad family A such that M F (A) adds a dominating real.
The classical properties of Menger and Hurewicz, and their generalizations, are defined in terms ... more The classical properties of Menger and Hurewicz, and their generalizations, are defined in terms of open covers of a topological space. The authors have introduced an abstract frame in which they develop these covering properties: a multicovered space is a set X with a family of covers of X. Natural examples of multicovered spaces appear in topological, metric and uniform spaces, and in topological groups. The authors extend the classical selection principles, including their game theoretic aspects, to this abstract setting, and give applications, in particular to topological groups. This paper is a survey of their results in this area.
Topology and its Applications, 2008
We introduce and study so-called C-separation properties leading to a fine hierarchy of spaces wi... more We introduce and study so-called C-separation properties leading to a fine hierarchy of spaces with the Hurewicz property fin (O, Γ). By definition, a topological space X has the C-separation property for a class C of spaces if for any embedding X ⊂ C into a space C ∈ C there is a σ-compact subset A ⊂ C containing X. It turns out that the classical Hurewicz property is equivalent to the G δ -separation property for the class G δ of Polish spaces. On the other extreme there are Sierpiński sets having the UMseparation property for the class UM of universally measurable spaces. We construct several examples distinguishing the C-separation properties for various descriptive classes C and also study the interplay between the C-separation properties and the selection principles fin (C, Γ).
Applied General Topology
In this paper we introduce and study three new cardinal topological invariants called the cs*, cs... more In this paper we introduce and study three new cardinal topological invariants called the cs*, cs-, and sb-characters. The class of topological spaces with countable cs*-character is closed under many topological operations and contains all aleph-spaces and all spaces with point-countable cs*-network. Our principal result states that each non-metrizable sequential topological group with countable cs*-character has countable pseudo-character and contains an open $k_\omega$-subgroup.
Topology and its Applications, 2010
Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topologica... more Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F (X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle S fin (O, Ω) (the latter means that for every sequence un n∈ω of open covers of T there exists a sequence vn n∈ω such that vn ∈ [un] <ω and for every F ∈ [X] <ω there exists n ∈ ω with F ⊂ ∪vn). This characterization gives a consistent answer to a problem posed by C. Hernandes, D. Robbie, and M.
The Journal of Symbolic Logic, 2008
Using a dictionary translating a variety of classical and modern covering properties into combina... more Using a dictionary translating a variety of classical and modern covering properties into combinatorial properties of continuous images, we get a simple way to understand the interrelations between these properties in ZFC and in the realm of the trichotomy axiom for upward closed families of sets of natural numbers. While it is now known that the answer to the Hurewicz 1927 problem is positive, it is shown here that semifilter trichotomy implies a negative answer to a slightly stronger form of this problem.
Journal of Pure and Applied Algebra, 2008
In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fré... more In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet-Urysohn fan S ω in a topological group G admitting a functorial embedding [0, 1] ⊂ G. The latter means that each autohomeomorphism of [0, 1] extends to a continuous homomorphism of G. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space X containing a topological copy of the space Q of rationals give topological groups containing S ω .
Archive for Mathematical Logic, 2011
We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-σ-bounded ... more We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-σ-bounded topological group G admits an increasing chain Gα : α < b(L) of its proper subgroups such that: (i) α Gα = G; and (ii) For every σ-bounded subgroup H of G there exists α such that H ⊂ Gα. In case of the group Sym(ω) of all permutations of ω with the topology inherited from ω ω this improves upon earlier results of S. Thomas.
Topology and its Applications, 2010
We study M -separability as well as some other combinatorial versions of separability. In particu... more We study M -separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b = d implies that the class of selectively separable spaces is not closed under finite products, even for the spaces of continuous functions with the topology of pointwise convergence. We also show that there exists no maximal M -separable countable space in the model of Frankiewicz, Shelah, and Zbierski in which all closed P -subspaces of ω * admit an uncountable family of nonempty open mutually disjoint subsets. This answers several questions of Bella, Bonanzinga, Matveev, and Tkachuk.
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Papers by Lyubomyr Zdomsky