arXiv subjects

István Juhász

Publications and source records attributed to István Juhász.

28 records · Page 2Linked to original sources

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

Variations on countable tightness

We consider 9 natural tightness conditions for topological spaces that are all variations on countable tightness and investigate the interrelationships between them. Several natural open problems are raised.

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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".

math.GN↗

arXiv2016

On $σ$-countably tight spaces

Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality $\mathfrak{c}$ if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and $σ$-countably tight compactum has cardinality $\mathfrak{c}$ remains open. We also show that if an arbitrary product is $σ$-countably tight then all but finitely many of its factors must be countably tight.

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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 $\,Δ(X)< {2^{{pd}(X)}}$.

math.GN↗

arXiv2014

Between countably compact and $ω$-bounded

Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. Here we study $P$-bounded spaces for the properties $P \in \{ωD, ωN, C_2 \}$ where $ωD \, \equiv$ "countable discrete", $ωN \, \equiv$ "countable nowhere dense", and $C_2 \,\equiv$ "second countable". Clearly, for each of these $P$-bounded is between countably compact and $ω$-bounded. We give examples in ZFC that separate all these boundedness properties and their appropriate combinations. Consistent separating examples with better properties (such as: smaller cardinality or weight, local compactness, first countability) are also produced. We have interesting results concerning $ωD$-bounded spaces which show that $ωD$-boundedness is much stronger than countable compactness: $\bullet$ Regular $ωD$-bounded spaces of Lindelöf degree $< cov(\mathcal{M})$ are $ω$-bounded. $\bullet$ Regular $ωD$-bounded spaces of countable tightness are $ωN$-bounded, and if $\mathfrak{b} > ω_1$ then even $ω$-bounded. $\bullet$ If a product of Hausdorff space is $ωD$-bounded then all but one of its factors must be $ω$-bounded. $\bullet$ Any product of at most $\mathfrak{t}$ many Hausdorff $ωD$-bounded spaces is countably compact. As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated.

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arXiv2010

Conflict free colorings of (strongly) almost disjoint set-systems

A set-system $X$ is a $(λ, κ,μ)$-system iff $|X|=λ$, $|x|=κ$ for each $x\in X$, and $X$ is $μ$-almost disjoint. We write $[λ, κ, μ] -> ρ$ iff every $(λ, κ,μ)$-system has a "conflict free coloring with $ρ$ colors", i.e. there is a coloring of the elements of $\cup X$ with$ρ$ colors such that for each element $x$ of $X$ there is a color $ξ<ρ$ such that exactly one element of $x$ has color $ξ$. Our main object of study is the relation $[λ, κ, μ] -> ρ$. We give full description of this relation when $κ$ is finite. We also show that if $d$ is a natural number then $[λ,κ,d]-> ω$ always holds. Under GCH we prove that $[λ,κ,ω]-> ω_2$ holds for $κ>ω_1$, but the relation $[λ,κ,ω]-> ω_1$ is independent (modulo some large cardinals).

math.LO↗

arXiv2004

Cardinal sequences and Cohen real extensions

We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most (2^{aleph_0})^V many levels of size omega. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of the regular and of the 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.

math.LO↗