arXiv subjects

Zoltán Szentmiklóssy

Publications and source records attributed to Zoltán Szentmiklóssy.

At least 19 recordsLinked to original sources

arXiv2025

Noetherian Properties, Large Cardinals, and Independence Around $\aleph_ω$

A base of a topological space is called {\em Noetherian } iff it does not contain an infinite strictly $\subseteq$-increasing chain. We show that minimal cardinality of a regular spaces without a Noetherian base is the first strongly inaccessible cardinal, answering a question from the 1980s. We also study the {\em Noetherian type} of a topological space $X$, denoted by $Nt(X)$, defined as the least cardinal $κ$ such that $X$ has a base $\mathcal B$ with $|\{B'\in \mathcal B: B\subset B'\}|<κ$ for each $B\in \mathcal B$. The behavior of the Noetherian type under the $G_δ$-modification was investigated by Milovich and Spadaro. A central question, posed by them, is whether the Noetherian type of the $G_δ$-modification of the space $D(2)^{\aleph_ω}$ is $ω_1$. This statement, denoted (Nt), is known to be independent of ZFC + GCH: it holds under ``GCH + $\square_{\aleph_ω}$'', but fails under ``GCH + $(\aleph_{ω+1}, \aleph_ω)\to (\aleph_1, \aleph_0)$''. We place this phenomenon in a broader context by identifying similar independence phenomena for several topological and combinatorial principles. These include: (wFN) the weak Freese-Nation property of $[\aleph_ω]^ω$; (SAT) the existence of a saturated MAD family in $[\aleph_ω]^ω$; (HnT) the existence of an $ω$-homogeneous, but not $ω$-transitive permutation group on $\aleph_ω$; and (SPL) the existence of a countably compact, locally countable, and $ω$-fair regular space of cardinality $\aleph_{ω+1}$. Assuming GCH, we analyze the logical relationships between these principles and show, for example, that SPL implies wFN, which in turn implies both SAT, HnT and Nt, while SAT does not imply Nt.

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arXiv2024

Pseudocompact versus countably compact in first countable spaces

The primary objective of this work is to construct spaces that are "pseudocompact but not countably compact," abbreviated as PNC, while endowing them with additional properties. First, motivated by an old problem of van Douwen, we construct from CH a locally compact and locally countable first countable PNC space with countable spread. A space is deemed "densely countably compact", denoted as DCC for brevity, if it possesses a dense, countable compact subspace. Moreover, a space qualifies as "densely relatively countably compact", abbreviated as DRC, if it contains a dense subset $D$ such that every infinite subset of $D$ has an accumulation point in $X$. A countably compact space is DCC, a DCC space is DRC, and a DRC space is evidently pseudocompact. The Tychonoff plank is a DCC space but is not countably compact. A $Ψ$-space belongs to the class of DRC spaces but is not DCC. Lastly, if $p\in ω^*$ is not a P-point, then $T(p)$, representing the type of $p$ in $ω^*$, constitutes a pseudocompact subspace of $ω^*$ that is not \DRC. When considering a topological property denoted as $Q$, we define a space $X$ as "hereditarily $Q$" if every regular closed subspace of $X$ also possesses property $Q$. The Tychonoff plank and the $Ψ$-spaces are not hereditary examples. However, the aforementioned space $T(p)$ is a hereditary example, albeit not being first countable. In this paper we want to find (first countable) examples which separates these properties hereditarily. We have obtained the following result. (1) There is a DCC space $X$ such that no $H\in RC(X)^+$ is countably compact. (2) If CH holds, then there is a DRC space $Y$ such that no $H\in RC(Y)^+$ is DCC. (3) If CH holds, then there is a first countable pseudocompact space $Z$ such that no $H\in RC(Z)^+$ is DRC.

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arXiv2024

The class $C(ω_1)$ and countable net weight

Hart and Kunen, and independently in the recent preprint arXiv:2304.13113, Ríos-Herrejón defined and studied the class $C(ω_1)$ of topological spaces $X$ having the property that for every neighborhood assignment $\{U(y) : y \in Y\}$ with $Y \in [X]^{ω_1}$ there is $Z \in [Y]^{ω_1}$ such that $$Z \subset \bigcap \{U(z) : z \in Z\}.$$ It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these and several other classes that are sandwiched between them. These clarify some of the main problems that were raised in the above preprint. In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in $C(ω_1)$ has countable net weight, answering a question that was raised by Hart and Kunen.

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arXiv2024

Selectively pseudocompact spaces

A novel selection principle was introduced by Dorantes-Aldama and Shakhmatov: a topological space $X$ is termed {\em selectively pseudocompact} if for any sequence $(U_n:n\in ω)$ of pairwise disjoint non-empty open sets of $X$, one can choose points $x_n\in U_n$ such that the sequence $(x_n:n\in ω)$ has an accumulation point. In this paper, we explore various versions of this principle when we permit the selection of finite, scattered, or nowhere dense sets instead of just singletons. We develop a method to prove that the aforementioned versions of selective pseudocompactness are indeed distinct from one another.

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arXiv2024

On the companion of spaces having dense, relatively countable compact subspaces

The notion of "pseudocompactness" was introduced by Hewitt. The concept of relatively countably compact subspaces were explored by Marjanovic to show that a $Ψ$-space is pseudocompact. A topological space is said to be DRC (DRS) iff it possesses a dense, relatively countably compact (or relatively sequentially compact, respectively) subspace. The concept of selectively pseudocompact game Sp(X) and the selectively sequentially pseudocompact game Ssp(X) were introduced by Dorantes-Aldama and Shakhmatov. They explored the relationship between the existence of a winning strategy and a stationary winning strategy for player P in these games. In particular, they observed that there exists a stationary winning strategy in the game Sp(X) (Ssp(X)) for Player P iff $X$ is DRC (or DRS, respectively). In this paper we introduce natural weakening of the properties DRC and DRS: a space $X$ is DRCo ( DRSo) iff there is a sequence $(D_n:n \in { ω})$ of dense subsets of $X$ such that every sequence $(d_n:n \in { ω} )$ with $d_n \in D_n$ has an accumulation point (or contains a convergent subsequence, respectively). These properties are also equivalent to the existence of some limited knowledge winning strategy on the corresponding games $Sp(X)$ and $Ssp(X)$. Clearly, DRS implies DRC and DRSo, DRC or DRSo imply DRCo. The main part of this paper is devoted to prove that apart from these trivial implications, consistently there are no other implications between these properties.

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arXiv2022

On resolvability of products

All spaces below are $T_0$ and crowded (i.e. have no isolated points). For $n \le ω$ let $M(n)$ be the statement that there are $n$ measurable cardinals and $Π(n)$ ($Π^+(n)$) that there are $n+1$ (0-dimensional $T_2$) spaces whose product is irresolvable. We prove that $M(1),\,Π(1)$ and $Π^+(1)$ are equiconsistent. For $1 < n < ω$ we show that $CON(M(n))$ implies $CON(Π^+(n))$. Finally, $CON(M(ω))$ implies the consistency of having infinitely many crowded 0-dimensional $T_2$-spaces such that the product of any finitely many of them is irresolvable. These settle old problems of Malychin. Concerning an even older question of Ceder and Pearson, we show that the following are consistent modulo a measurable cardinal: (i) There is a 0-dimensional $T_2$ space $X$ with $ω_2 \le Δ(X) \le 2^{ω_1}$ whose product with any countable space is not $ω_2$-resolvable, hence not maximally resolvable. (ii) There is a monotonically normal space $X$ with $Δ(X) = \aleph_ω$ whose product with any countable space is not $ω_1$-resolvable, hence not maximally resolvable. These significantly improve a result of Eckertson.

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arXiv2021

Large strongly anti-Urysohn spaces exist

As defined in [1], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any two infinite} closed subsets of it intersect. Our main result answers the two main questions of [1] by providing a ZFC construction of a locally countable SAU space of cardinality $2^{\mathfrak{c}}$. The construction hinges on the existence of $2^{\mathfrak{c}}$ weak P-points in $ω^*$, a very deep result of Ken Kunen. It remains open if SAU spaces of cardinality $> 2^{\mathfrak{c}}$ could exist, while it was shown in [1] that $2^{2^{\mathfrak{c}}}$ is an upper bound. Also, we do not know if crowded SAU spaces, i.e. ones without any isolated points, exist in ZFC but we obtained the following consistency results concerning such spaces. (1) It is consistent that $\mathfrak{c}$ is as large as you wish and there is a locally countable and crowded SAU space of cardinality $\mathfrak{c}^+$. (2) It is consistent that both $\mathfrak{c}$ and $2^\mathfrak{c}$ are as large as you wish and there is a crowded SAU space of cardinality $2^\mathfrak{c}$. (3) For any uncountable cardinal $κ$ the following statements are equivalent: (i) $κ=cof({[κ]}^{ω},\subseteq)$. (ii) There is a locally countable and crowded SAU space of size $κ$ in the generic extension obtained by adding $κ$ Cohen reals. (iii) There is a locally countable and countably compact $T_1$-space of size $κ$ in some CCC generic extension. [1] I. Juhasz, L. Soukup, and Z. Szentmiklossy, Anti-Urysohn spaces, Top. Appl., 213 (2016), pp. 8--23.

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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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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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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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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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arXiv2016

First countable and almost discretely Lindelöf $T_3$ spaces have cardinality at most continuum

A topological space $X$ is called almost discretely Lindelöf if every discrete set $D \subset X$ is included in a Lindelöf subspace of $X$. We say that the space $X$ is {\em $μ$-sequential} if for every non-closed set $A \subset X$ there is a sequence of length $\le μ$ in $A$ that converges to a point which is not in $A$. With the help of a technical theorem that involves elementary submodels, we establish the following two results concerning such spaces. (1) For every almost discretely Lindelöf $T_3$ space $X$ we have $|X| \le 2^{χ(X)}$. (2) If $X$ is a $μ$-sequential $T_2$ space of pseudocharacter $ψ(X) \le 2^μ$ and for every free set $D \subset X$ we have $L(\overline{D}) \le μ$, then $|X| \le 2^μ$. The case $χ(X) = ω$ of (1) provides a solution to Problem 4.5 from "I. Juhász, V. Tkachuk, and R. Wilson, Weakly linearly Lindelöf monotonically normal spaces are Lindelöf", while the case $μ= ω$ of (2) is a partial improvement on the main result of "A.V. Archangel'skii and R.Z. Buzyakova, On some properties of linearly Lindelöf spaces".

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arXiv2015

Anti-Urysohn spaces

All spaces are assumed to be infinite Hausdorff spaces. We call a space "anti-Urysohn" $($AU in short$)$ iff any two non-emty regular closed sets in it intersect. We prove that $\bullet$ for every infinite cardinal $κ$ there is a space of size $κ$ in which fewer than $cf(κ)$ many non-empty regular closed sets always intersect; $\bullet$ there is a locally countable AU space of size $κ$ iff $ω\le κ\le 2^{\mathfrak c}$. A space with at least two non-isolated points is called "strongly anti-Urysohn" $($SAU in short$)$ iff any two infinite closed sets in it intersect. We prove that $\bullet$ if $X$ is any SAU space then $ \mathfrak s\le |X|\le 2^{2^{\mathfrak c}}$; $\bullet$ if $\mathfrak r=\mathfrak c$ then there is a separable, crowded, locally countable, SAU space of cardinality $\mathfrak c$; \item if $λ> ω$ Cohen reals are added to any ground model then in the extension there are SAU spaces of size $κ$ for all $κ\in [ω_1,λ]$; $\bullet$ if GCH holds and $κ\leλ$ are uncountable regular cardinals then in some CCC generic extension we have $\mathfrak s=κ$, $\,\mathfrak c=λ$, and for every cardinal $μ\in [\mathfrak s, \mathfrak c]$ there is an SAU space of cardinality $μ$. The questions if SAU spaces exist in ZFC or if SAU spaces of cardinality $> \mathfrak c$ can exist remain open.

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arXiv2015

Pinning Down versus Density

The pinning down number $ {pd}(X)$ of a topological space $X$ is the smallest cardinal $κ$ such that for any neighborhood assignment $U:X\to τ_X$ there is a set $A\in [X]^κ$ with $A\cap U(x)\ne\emptyset$ for all $x\in X$. Clearly, c$(X) \le {pd}(X) \le {d}(X)$. Here we prove that the following statements are equivalent: (1) $2^κ<κ^{+ω}$ for each cardinal $κ$; (2) ${d}(X)={pd}(X)$ for each Hausdorff space $X$; (3) ${d}(X)={pd}(X)$ for each 0-dimensional Hausdorff space $X$. This answers two questions of Banakh and Ravsky. The dispersion character $Δ(X)$ of a space $X$ is the smallest cardinality of a non-empty open subset of $X$. We also show that if ${pd}(X)<{d}(X)$ then $X$ has an open subspace $Y$ with ${pd}(Y)<{d}(Y)$ and $|Y| = Δ(Y)$, moreover the following three statements are equiconsistent: (i) There is a singular cardinal $λ$ with $pp(λ)>λ^+$, i.e. Shelah's Strong Hypothesis fails; (ii) there is a 0-dimensional Hausdorff space $X$ such that $|X|=Δ(X)$ is a regular cardinal and ${pd}(X)<{d}(X)$; (iii) there is a topological space $X$ such that $|X|=Δ(X)$ is a regular cardinal and ${pd}(X)<{d}(X)$. We also prove that $\bullet$ ${d}(X)={pd}(X)$ for any locally compact Hausdorff space $X$; $\bullet$ for every Hausdorff space $X$ we have $|X|\le 2^{2^{{pd}(X)}}$ and ${pd}(X)<{d}(X)$ implies $Δ(X)< 2^{2^{{pd}(X)}}$; $\bullet$ for every regular space $X$ we have $\min\{Δ(X),\, w(X)\}\le 2^{{pd}(X)}\,$ and ${d}(X)<2^{{pd}(X)},\,$ moreover ${pd}(X)<{d}(X)$ implies $\,Δ(X)< {2^{{pd}(X)}}$.

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