arXiv subjects
Publications and source records attributed to Lyubomyr Zdomskyy.
We introduce a density counterpart of the Scheepers covering property $\bigcup_{\mathrm{fin}}(\mathcal O,Ω)$ and study its relations to known combinatorial density property. In particular, we show that it is equivalent to the $M$-separability under the Near Coherence of Filters principle of Blass and Weiss.
A set of reals $X$ is $\mathfrak{b}$-concentrated if it has cardinality at least $\mathfrak{b}$ and it contains a countable set $D\subseteq X$ such that each closed subset of $X$ disjoint with $D$ has size smaller than $\mathfrak{b}$. We present ZFC results about structures of $\mathfrak{b}$-concentrated sets with the Hurewicz covering property using semifilters. Then we show that assuming that the semifilter trichotomy holds, then each $\mathfrak{b}$-concentrated set is Hurewicz and even productively Hurewicz. We also show that the appearance of Hurewicz $\mathfrak{b}$-concentrated sets under the semifilter trichotomy is somewhat specific and the situation in the Laver model for the consitency of the Borel Conjecture is different.
We provide examples of spaces satisfying generalized combinatorial covering properties such as the Hurewicz, Menger, and $γ$-properties in an uncountable setting. Our approach is motivated by canonical constructions from the classical countable case, including the examples of Bartoszyński and Shelah separating the Hurewicz property from $σ$-compactness, the examples of Tsaban and Zdomskyy separating the Hurewicz and Menger properties, and Tsaban's construction of a nontrivial set of reals with the $γ$-property. We focus on the genuinely nontrivial aspects of these higher-cardinal generalizations, uncovering several open problems whose nature appears substantially different from that of their countable counterparts.
We prove that consistently $ω_1<\mathfrak{c}$ and there exists a compact space $K$ whose Banach space $C(K)$ of continuous real-valued functions is Grothendieck, has density $ω_1$, and admits no renorming which is strictly convex or sequentially Kadets--Klee.
We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped continuously onto the Cantor space. These results allow to separate the properties of Menger and $\mathsf{S}_1(Γ,\mathrm{O})$ in the realm of sets of reals without perfect subsets and solve a problem of Nowik and Tsaban concerning perfectly meager subsets in the transitive sense. We present also some other applications of the mentioned above methods.
We prove that assuming $\mathfrak{b}=\mathfrak{d}$, in the class of hereditarily Lindelöf spaces, each productively Scheepers space is productively Hurewicz. The above statement remains true in the class of all general topological spaces assuming that $\mathfrak{d}=\aleph_1$. To this end we use combinatorial methods and the Menger covering property parametrized by ultrafilters. We also show that if near coherence of filters holds, then the Scheepers property is equivalent to a Menger property parametrized by any ultrafilter.
We work in the realm of sets of reals. We prove that in the Miller model and in a model constructed by Goldstern-Judah-Shelah all universally meager sets have size at most $ω_1$. Some relations between combinatorial covering properties in these models allow to obtain the same limitations for sizes of Rothberger spaces and Hurewicz spaces with no homeomorphic copy of the Cantor set inside. It follows from our results that the existence of a strong measure zero set of size $ω_2$ does not imply the existence of a Rothberger space of size $ω_2$. We also prove that in the Miller model all strong measure zero sets have size at most $ω_1$.
Using iterated Sacks forcing and topological games, we prove that the existence of a totally imperfect Menger set in the Cantor cube with cardinality continuum is independent from ZFC. We also analyze the structure of Hurewicz and consonant subsets of the Cantor cube in the Sacks model.
Given a graph $G$ and a subset $X$ of vertices of $G$ with size at least two, we denote by $N^2_G(X)$ the set of vertices of $G$ that have at least two neighbors in $X$. We say that a bipartite graph $G$ with sides $A$ and $B$ satisfies the double Hall property if for every subset $X$ of vertices of $A$ with size at least 2, $\vert N^2_G(X)\vert \geq \vert X\vert$. Salia conjectured that if $G$ is a bipartite graph that satisfies the double Hall property, then there exists a cycle in $G$ that covers all vertices of $A$. In this work, we study this conjecture restricted to infinite graphs. For this, we use the definition of ends and infinite cycles. It is simple to see that Salia's conjecture is false for infinite graphs in general. Consequently, all our results are partial. Under certain hypothesis it is possible to obtain a collection of pairwise disjoint 2-regular subgraphs that covers $A$. We show that if side $B$ is locally finite and side $A$ is countable, then the conjecture is true. Furthermore, assuming the conjecture holds for finite graphs, we show that it holds for infinite graphs with a restriction on the degree of the vertices of $B$. This result is inspired by the result obtained by Barát, Grzesik, Jung, Nagy and Pálvölgyi for finite graphs. Finally, we also show that if Salia's conjecture holds for some cases of infinite graphs, then the conjecture about finite graphs presented by Lavrov and Vandenbussche is true.
We consider products of sets of reals with a combinatorial structure based on scales parameterized by filters. This kind of sets were intensively investigated in products of spaces with combinatorial covering properties as Hurewicz, Scheepers, Menger and Rothberger. We will complete this picture with focusing on properties from the second row of the Scheepers diagram. In particular we show that in the Miller model a product space of two $\mathfrak{d}$-concentrated sets has a strong covering property $\mathsf{S}_1(Γ,Ω)$. We provide also counterexamples in products to demonstrate limitations of used methods.
This article is devoted to the interplay between productively Menger and productively Hurewicz subspaces of the Cantor space. In particular, we show that in the Laver model for the consistency of the Borel's conjecture these two notions coincide and characterize Hurewicz spaces. On the other hand, it is consistent with CH that there are productively Hurewicz subspaces of the Cantor space which are not productively Menger.
We prove that it is consistent with ZFC that for every non-decreasing function $f:[0,1]\to [0,1]$, each subset of $[0,1]$ of cardinality $\mathfrak c$ contains a set of cardinality $\mathfrak c$ on which $f$ is uniformly continuous. We show that this statement follows from the assumptions that $\mathfrak d^* < \mathfrak c$ and $\mathfrak c$ is regular, where $\mathfrak d^*\leq \mathfrak d$ is the smallest cardinality $κ$ such that any two disjoint countable dense sets in the Cantor set can be separated by sets each of which is an intersection of at most $κ$-many open sets in the Cantor set. We establish also that $\mathfrak d^*=\min\{\mathfrak u, \mathfrak d\}=\min\{\mathfrak r, \mathfrak d\}$, thus giving an alternative proof of the latter equality established by J. Aubrey in 2004.
Building on results of Medvedev, we construct a $\mathsf{ZFC}$ example of a non-Polish topological group that is countable dense homogeneous. Our example is a dense subgroup of $\mathbb{Z}^ω$ of size $\mathfrak{b}$ that is a $λ$-set. We also conjecture that every countable dense homogenous Baire topological group with no isolated points contains a copy of the Cantor set, and give a proof in a very special case.
In this paper, we show that the existence of certain first-countable compact-like extensions is equivalent to the equality between corresponding cardinal characteristics of the continuum. For instance, $\mathfrak b=\mathfrak s=\mathfrak c$ if and only if every regular first-countable space of weight $< \mathfrak c$ can be densely embedded into a regular first-countable countably compact space.
In this paper we investigate R-,H-, and M-{\it nw}-selective properties introduced in \cite{BG}. In particular, we provide consistent uncountable examples of such spaces and we define \textit{trivial} R-,H-, and M-{\it nw}-selective spaces the ones with countable net weight having, additionally, the cardinality and the weight strictly less then $cov({\cal M})$, $\frak b$, and $\frak d$, respectively. Since we establish that spaces having cardinalities more than $cov({\cal M})$, $\frak b$, and $\frak d$, fail to have the R-,H-, and M-{\it nw}-selective properties, respectively, non-trivial examples should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R-{\it nw}-selective and H-{\it nw}-selective spaces and we establish some limitations to constructions of non-trivial examples. Moreover, we consistently prove the existence of two H-{\it nw}-selective spaces whose product fails to be M-{\it nw}-selective. Finally, we study some relations between {\it nw}-selective properties and a strong version of the HFD property.
In this paper, we investigate the poset $\mathbf{OF}(X)$ of free open filters on a given space $X$. In particular, we characterize spaces for which $\mathbf{OF}(X)$ is a lattice. For each $n\in\mathbb{N}$ we construct a scattered space $X$ such that $\mathbf{OF}(X)$ is order isomorphic to the $n$-element chain, which implies the affirmative answer to two questions of Mooney. Assuming CH we construct a scattered space $X$ such that $\mathbf{OF}(X)$ is order isomorphic to $(ω+1,\geq)$. To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of $β(κ)$. Assuming the existence of $n$ measurable cardinals, for every $m_0,\ldots,m_{n}\in\mathbb N$ we construct a space $X$ such that $\mathbf{OF}(X)$ is order isomorphic to $\prod_{i=0}^nm_i$. Also, we show that the existence of a metric space possessing a free $ω_1$-complete closed, $G_δ$, $F_σ$ or Borel ultrafilter is equivalent to the existence of a measurable cardinal.
Using combinatorial covering properties, we show that there is no concentrated set of reals of size $ω_2$ in the Miller model. The main result refutes a conjecture of Bartoszyński and Halbeisen. We also prove that there are no $γ$-set of reals of size $ω_2$ in the Miller model.
Improving a result of M. Talagrand, under the assumption of a weak form of Martin's axiom, we construct a totally disconnected compact Hausdorff space $K$ such that the Banach space $C(K)$ of continuous real-valued functions on $K$ is a Grothendieck space but there exists a sequence $(μ_n)$ of Radon measures on $K$ such that $μ_n(A)\to0$ for every clopen set $A\subseteq K$ and $\int_Kfdμ_n\not\to0$ for some $f\in C(K)$. Consequently, we get that Martin's axiom implies the existence of a Boolean algebra with the Grothendieck property but without the Nikodym property.