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Lajos Soukup

Publications and source records attributed to Lajos Soukup.

At least 37 records · Page 2Linked to original sources

arXiv2021

On a problem of Angelo Bella

The main result of this note is the following theorem. "If $X$ is any Hausdorff space with $κ= \widehat{F}(X) \cdot \widehatμ(X)$ then $L(X_{< κ}) \le \varrho(κ)$". Here $\widehat{F}(X)$ is the smallest cardinal $φ$ so that $|S| < φ$ for any set $S$ that is free in $X$ and $\widehatμ(X)$ is the smallest cardinal $μ$ so that, for every set $S$ that is free in $X$, any open cover of $\overline {S}$ has a subcover of size $< μ$. Moreover, $X_{< κ}$ is the $G_{< κ}$-modification of $X$ and $\varrho(κ) = \min \{\varrho : \varrho ^{< κ} = \varrho\}$. As a corollary we obtain that if $X$ is a linearly Lindelöf regular space of countable tightness then $L(X_δ) \le \mathfrak{c}$, provided that $ \mathfrak{c} = 2^{< \mathfrak{c}}$. This yields a consistent affirmative answer to a question of Angelo Bella.

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arXiv2021

The double density spectrum of a topological space

It is an interesting, maybe surprising, fact that different dense subspaces of even "nice" topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological space $X$ that we call the double density spectrum of $X$ and denote by $dd(X)$. We improve a result of Berner and Juhasz by showing that $dd(X)$ is always $ω$-closed (i.e. countably closed) if $X$ is Hausdorff. We manage to give complete characterizations of the double density spectra of Hausdorff and of regular spaces as follows. Let $S$ be a non-empty set of infinite cardinals. Then (1) $S = dd(X)$ holds for a Hausdorff space $X$ iff S is $ω$-closed and $sup S \le 2^{2^{\min S}},$ (2) S = dd(X) holds for a regular space X iff S is $ω$-closed and $\sup S \le {2^{\min S}}$. We also prove a number of consistency results concerning the double density spectra of compact spaces. For instance: (i) If $κ= cf(κ)$ embeds in $\mathcal{P}(ω)/fin$ and $S$ is any set of uncountable regular cardinals $< κ$ with $|S| < \min S$, then there is a compactum $C$ such that $\{ω, κ\} \cup S \subset dd(C)$, moreover $λ\notin d(C)$ whenever $|S| + ω< cf(λ) < κ$ and $cf(λ) \notin S$. (ii) It is consistent to have a separable compactum $C$ such that $dd(C)$ is not $ω_1$-closed.

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arXiv2021

The mixing time of the switch Markov chains: a unified approach

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms. The aim of this paper is to unify these approaches. We will illustrate the strength of the unified method by showing that on any $P$-stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the switch Markov chain on a region of degree sequences. Two applications of this general result will be presented. One is an almost uniform sampler for power-law degree sequences with exponent $γ>1+\sqrt{3}$. The other one shows that the switch Markov chain on the degree sequence of an Erdős-Rényi random graph $G(n,p)$ is asymptotically almost surely rapidly mixing if $p$ is bounded away from 0 and 1 by at least $\frac{5\log n}{n-1}$.

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arXiv2021

Minimal vertex covers in infinite hypergraphs

In this paper a hypergraph will be identified with the family of its edges. A hypergraph $\mathcal E$ possesses property $C(k,ρ)$ iff $|\bigcap \mathcal E'|<ρ$ for each $\mathcal E'\in {[\mathcal E]}^{k}$. A vertex set $Y\subset \bigcup\mathcal E$ is a "vertex cover" of $\mathcal E$ iff $E\cap Y\ne \emptyset$ for each $E\in \mathcal E$. A vertex cover $Y$ is "minimal" iff no proper subset of $Y$ is vertex cover. If $A$ is a set and $S$ is a set of cardinals, write $$ {[A]}^{S}=\{B\subset A: |B|\in S\}. $$ If $λ$ and $ρ$ are cardinals, $S$ is a set of cardinals, $k\in ω$, then we write $$\mathbf M({λ},{S},{k},{μ})\to \mathbf{MinVC} $$ iff every hypergraph $\mathcal E\subset {[λ]}^{S}$ possessing property $C({k},{ρ})$ has a minimal vertex cover. If $S=\{κ\}$, then we simply write $\mathbf M({λ},κ,{k},{μ})\to \mathbf{MinVC}$ for $\mathbf M({λ},\{κ\},{k},{μ})\to \mathbf{MinVC}$ A set $S$ of cardinals is "nowhere stationary" iff $S\cap α$ is not stationary in $α$ for any ordinal $α$ with $cf(α)>ω$. Countable sets of cardinals, and sets of successor cardinals are nowhere stationary. In this paper we prove: (1) $\mathbf M({λ},{S},{2},{k})\to \mathbf{MinVC}$ for each nowhere stationary set $S$ of cardinals and $ω\le λ$, (2) $\mathbf M({λ},{κ} ,{2},{ρ})\to \mathbf{MinVC}$ provided $ρ<\beth_ω\le κ\le λ$, (3) $\mathbf M({λ},{ω},{r},{k})\to \mathbf{MinVC}$ provided $ω\le λ$ and $k,r\in ω$, (4) $\mathbf M({λ},{ω_1},{3},{k})\to \mathbf{MinVC}$ provided $ω_1\le λ$ and $k\in ω$.

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arXiv2020

Dominating and pinning down pairs for topological spaces

We call a pair of infinite cardinals $(κ,λ)$ with $κ> λ$ a dominating (resp. pinning down) pair for a topological space $X$ if for every subset $A$ of $X$ (resp. family $\mathcal{U}$ of non-empty open sets in $X$) of cardinality $\le κ$ there is $B \subset X$ of cardinality $\le λ$ such that $A \subset \overline{B}$ (resp. $B \cap U \ne \emptyset$ for each $U \in \mathcal{U}$). Clearly, a dominating pair is also a pinning down pair for $X$. Our definitions generalize the concepts introduced in [GTW] resp. [BT] which focused on pairs of the form $(2^λ,λ)$. The main aim of this paper is to answer a large number of the numerous problems from [GTW] and [BT] that asked if certain conditions on a space $X$ together with the assumption that $(2^λ,λ)$ or $((2^λ)^+,λ)$ is a pinning down pair or \dominating pair for $X$ would imply $d(X) \le λ$. [BT] A. Bella, V.V. Tkachuk, Exponential density vs exponential domination, preprint [GTW] G. Gruenhage, V.V. Tkachuk, R.G. Wilson, Domination by small sets versus density, Topology and its Applications 282 (2020)

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arXiv2020

On the free set number of topological spaces and their $G_δ$-modifications

For a topological space $X$ we propose to call a subset $S \subset X$ "free in $X$" if it admits a well-ordering that turns it into a free sequence in $X$. The well-known cardinal function $F(X)$ is then definable as $\sup\{|S| : S \text{ is free in } X\}$ and will be called the free set number of $X$. We prove several new inequalities involving $F(X)$ and $F(X_δ)$, where $X_δ$ is the $G_δ$-modification of $X$: $\bullet$ $L(X) \le 2^{2^{F(X)}}$ if $X$ is $T_2$ and $L(X)\le 2^{F(X)}$ if $X$ is $T_3$; $\bullet$ $|X|\le 2^{2^{F(X) \cdot ψ_c(X)}} \le 2^{2^{F(X) \cdot χ(X)}}$ for any $T_2$-space $X$; $\bullet$ $F(X_δ)\le 2^{2^{2^{F(X)}}}$ if $X$ is $T_2$ and $F(X_δ)\le 2^{2^{F(X)}}$ if $X$ is $T_3$.

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arXiv2020

On $κ$-homogeneous, but not $κ$-transitive permutation groups

A permutation group $G$ on a set $A$ is $κ$-homogeneous iff for all $X,Y\in [A]^κ$ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. $G$ is $κ$-transitive iff for any injective function $f$ with $dom(f)\cup ran(f)\in [A]^{\le κ}$ and $|A\setminus dom(f)|=|A\setminus ran(f)|=|A|$ there is a $g\in G$ with $f\subset g$. Giving a partial answer to a question of P. M. Neumann we show that there is an $ω$-homogeneous but not $ω$-transitive permutation group on a cardinal $λ$ provided (i) $λ<ω_ω$, or (ii) $2^ω<λ$, and $μ^ω=μ^+$ and $\Box_μ$ hold for each $μ\leλ$ with $ω=cf(μ)<{μ}$, or (iii) our model was obtained by adding $ω_1$ many Cohen generic reals to some ground model. For $κ>ω$ we give a method to construct large $κ$-homogeneous, but not $κ$-transitive permutation groups. Using this method we show that there exists $κ^+$-homogeneous, but not $κ^+$-transitive permutation groups on $κ^{+n}$ for each infinite cardinal $κ$ and natural number $n\ge 1$ provided $V=L$.

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arXiv2019

On first countable, cellular-compact spaces

As it was introduced by Tkachuk and Wilson, a topological space $X$ is cellular-compact if given any cellular, i.e. disjoint, family $\mathcal U$ of non-empty open subsets of $X$ there is a compact subspace $K\subset X$ such that $K\cap U\ne \emptyset$ for each $U\in \mathcal U$. Answering several questions raised by Tkachuk and Wilson we show that (1) any first countable cellular-compact $T_2$ space is $T_3$, and so its cardinality is at most $\mathfrak{c} = 2^ω$; (2) $cov(\mathcal M)>ω_1$ implies that every first countable and separable cellular-compact $T_2$ space is compact; (3 if there is no $S$-space then any cellular-compact $T_3$ space of countable spread is compact; (4) $MA_{ω_1}$ implies that every point of a compact $T_2$ space of countable spread has a disjoint local $π$-base.

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arXiv2019

Spaces of small cellularity have nowhere constant continuous images of small weight

We call a continuous map $f : X \to Y$ nowhere constant if it is not constant on any non-empty open subset of its domain $X$. Clearly, this is equivalent with the assumption that every fiber $f^{-1}(y)$ of $f$ is nowhere dense in $X$. We call the continuous map $f : X \to Y$ pseudo-open if for each nowhere dense $Z \subset Y$ its inverse image $f^{-1}(Z)$ is nowhere dense in $X$. Clearly, if $Y$ is crowded, i.e. has no isolated points, then $f$ is nowhere constant. The aim of this paper is to study the following, admittedly imprecise, question: How "small" nowhere constant, resp. pseudo-open continuous images can "large" spaces have? Our main results yield the following two precise answers to this question, explaining also our title. Both of them involve the cardinal function $\widehat{c}(X)$, the "hat version" of cellularity, which is defined as the smallest cardinal $κ$ such that there is no $κ$-sized disjoint family of open sets in $X$. Thus, for instance, $\widehat{c}(X) = ω_1$ means that $X$ is CCC. THEOREM A. Any crowded Tychonov space $X$ has a crowded Tychonov nowhere constant continuous image $Y$ of weight $w(Y) \le \widehat{c}(X)$. Moreover, in this statement $\le$ may be replaced with $<$ iff there are no $\widehat{c}(X)$-Suslin lines (or trees). THEOREM B. Any crowded Tychonov space $X$ has a crowded Tychonov pseudo-open continuous image $Y$ of weight $w(Y) \le 2^{<\widehat{c}(X)}$. If Martin's axiom holds then there is a CCC crowded Tychonov space $X$ such that for any crowded Hausdorff pseudo-open continuous image $Y$ of $X$ we have $w(Y) \ge \mathfrak{c}\,( = 2^{< ω_1})$.

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arXiv2019

A consistency result on long cardinal sequences

For any regular cardinal $κ$ and ordinal $η<κ^{++}$ it is consistent that $2^κ$ is as large as you wish, and every function $f:η\to [κ,2^κ]\cap Card$ with $f(α)=κ$ for $cf(α)<κ$ is the cardinal sequence of some locally compact scattered space.

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arXiv2018

On cardinal sequences of length < omega3

We prove the following consistency result for cardinal sequences of length $< \om_3$: if GCH holds and $\la \geq \om_2$ is a regular cardinal, then in some cardinal-preserving generic extension $2^{\om} = \la$ and for every ordinal $η< \om_3$ and every sequence $f = \langle \ka_{\al} : \al < η\rangle$ of infinite cardinals with $\ka_{\al}\leq \la$ for $\al < η$ and $\ka_{\al} = \om$ if $\mbox{cf}(\al) = \om_2$, we have that $f$ is the cardinal sequence of some LCS space. Also, we prove that for every specific uncountable cardinal $λ$ it is relatively consistent with ZFC that for every $\al,\be < \om_3$ with $\mbox{cf}(\al) < \om_2$ there is an LCS space $Z$ such that $\mbox{CS}(Z) = \langle ω\rangle_α\concat \langle λ\rangle_β$.

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arXiv2018

Left-Separating Order Types

A well ordering < of a topological space X is "left-separating" if $\{x'\in X: x'< x\}$ is closed in X for any x in X. A space is "left-separated" if it has a left-separating well-ordering. The left-separating type, $ord_l(X)$, of a left-separated space X is the minimum of the order types of the left-separating well orderings of X. We prove that (1) if $κ$ is a regular cardinal, then for each ordinal $α<κ^+$ there is a $T_2$ space $X$ with $ord_l(X)=κ\cdot α$; (2) if $κ=λ^+$ and $cf(λ)=λ>ω$, then for each ordinal $α<κ^+$ there is a 0-dimensional space $X$ with $ord_l( X)=κ\cdot α$; (3) if $κ=2^ω$ or $κ=\beth_{β+1}$, where $cf(β)=ω$, then for each ordinal $α<κ^+$ there is a locally compact, locally countable, 0-dimensional space $X$ with $ord_l( X)=κ\cdot α$. The union of two left-separated spaces is not necessarily left-separated. We show, however, that if X is a countably tight space, $X=Y\cup Z, ord_l(Y)$, $ord_l(Z)<ω_1 \cdot ω$, then $X$ is also left-separated and $ord_l(X)\le ord_l(Y)+ord_l(Z)$. We prove that it is consistent that there is a first countable, 0-dimensional space X, which is not left-separated, but there is a c.c.c poset Q such that in the generic extension $V^Q$ we have $ord_l(X)=ω_1 \cdot ω$. However, if $X$ is a topological space and $Q$ is a c.c.c poset such that in in the generic extension $V^Q$ we have $ord_l(X)<ω_1 \cdot ω$ then X is left-separated even in $V$.

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arXiv2018

On the tightness of $G_δ$-modifications

The $G_δ$-modification $X_δ$ of a topological space $X$ is the space on the same underlying set generated by, i.e. having as a basis, the collection of all $G_δ$ subsets of $X$. Bella and Spadaro recently investigated the connection between the values of various cardinal functions taken on $X$ and $X_δ$, respectively. In their paper, as Question 2, they raised the following problem: Is $t(X_δ) \le 2^{t(X)}$ true for every (compact) $T_2$ space $X$? Note that this is actually two questions. In this note we answer both questions: In the compact case affirmatively and in the non-compact case negatively. In fact, in the latter case we even show that it is consistent with ZFC that no upper bound exists for the tightness of the $G_δ$-modifications of countably tight, even Frechet spaces.

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arXiv2018

On the resolvability of Lindelöf-generated and (countable extent)-generated spaces

Given a topological property $P$, we say that the space $X$ is $P$-generated if for any subset $A\subset X$ that is not open in $X$ there is a subspace $Y \subset X$ with property $P$ such that $A\cap Y$ is not open in $Y$. (Of course, in this definition we could replace "open" with "closed".) In this paper we prove the following two results: (1) Every Lindelöf-generated regular space $X$ satisfying $|X|=Δ(X)=ω_1$ is $ω_1$-resolvable. (2) Any (countable extent)-generated regular space $X$ satisfying $Δ(X)>ω$ is $ω$-resolvable. These are significant strengthenings of our earlier results from [JSSz] which can be obtained from (1) and (2) by simply omitting the "-generated" part. Moreover, the second result improves a recent result of Filatova and Osipov from [FO] which states that Lindelöf-generated regular spaces of uncountable dispersion character are 2-resolvable. [FO] Maria A. Filatova, Alexander V. Osipov On resolvability of Lindelöf generated spaces, arxiv:1712.00803. Siberian Electronic Mathematical Reports, Vol. 14, (2017) pp. 1444-1444. [JSSz] Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán, Regular spaces of small extent are $ω$-resolvable. Fund. Math. 228 (2015), no. 1, 27-46.

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arXiv2018

Infinite combinatorics plain and simple

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

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arXiv2017

Coloring Cantor sets and resolvability of pseudocompact spaces

Let us denote by $Φ(λ,μ)$ the statement that $\mathbb{B}(λ) = D(λ)^ω$, i.e. the Baire space of weight $λ$, has a coloring with $μ$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(λ)$ picks up all the $μ$ colors. We call a space $X\,$ {\em $π$-regular} if it is Hausdorff and for every non-empty open set $U$ in $X$ there is a non-empty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called {\em feebly compact} if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact iff it is feebly compact. The main result of this paper is the following. Theorem. Let $X$ be a crowded feebly compact $π$-regular space and $μ$ be a fixed (finite or infinite) cardinal. If $Φ(λ,μ)$ holds for all $λ< \widehat{c}(X)$ then $X$ is $μ$-resolvable, i.e. contains $μ$ pairwise disjoint dense subsets. (Here $\widehat{c}(X)$ is the smallest cardinal $κ$ such that $X$ does not contain $κ$ many pairwise disjoint open sets.) This significantly improves earlier results of van Mill , resp. Ortiz-Castillo and Tomita.

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arXiv2017

Connected and/or topological group pd-examples

The pinning down number $pd(X)$ of a topological space $X$ is the smallest cardinal $κ$ such that for every neighborhood assignment $\mathcal{U}$ on $X$ there is a set of size $κ$ that meets every member of $\mathcal{U}$. Clearly, $pd(X) \le d(X)$ and we call $X$ a pd-example if $pd(X) < d(X)$. We denote by $\mathbf{S}$ the class of all singular cardinals that are not strong limit. It was proved in a paper of Juhász,Soukup and Szentmiklóssy (arXiv:1506.00206}) that TFAE: (1) $\mathbf{S} \ne \emptyset$; (2) there is a 0-dimensional $T_2$ pd-example; (3) there is a $T_2$ pd-example. The aim of this paper is to produce pd-examples with further interesting topological properties like connectivity or being a topological group by presenting several constructions that transform given pd-examples into ones with these additional properties. We show that $\mathbf{S} \ne \emptyset$ is also equivalent to the existence of a connected and locally connected $T_3$ pd-example, as well as to the existence of an abelian $T_2$ topological group pd-example. However, $\mathbf{S} \ne \emptyset$ in itself is not sufficient to imply the existence of a connected $T_{3.5}$ pd-example. But if there is $μ\in \mathbf{S}$ with $μ\ge \mathfrak{c}$ then there is an abelian $T_2$ topological group (hence $T_{3.5}$) pd-example which is also arcwise connected and locally arcwise connected. Finally, the same assumption $\,\mathbf{S} \setminus \mathfrak{c} \ne \emptyset\,$ even implies that there is a locally convex topological vector space pd-example.

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arXiv2017

How to drive our families mad

Given a family $F$ of pairwise almost disjoint sets on a countable set $S$, we study maximal almost disjoint (mad) families $F^+$ extending $F$. We define $a^+(F)$ to be the minimal possible cardinality of $F^+\setminus F$ for such $F^+$, and $a^+(κ)=\sup\{a^+(F): |F| \leq κ\}$. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\aleph_1=a<a^+(\aleph_1)=c$ and $a=a^+(\aleph_1)<c$ are both consistent. We also give a several constructions of mad families with some additional properties.

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