arXiv subjects

Istvan Juhasz

Publications and source records attributed to Istvan Juhasz.

17 recordsLinked to original sources

arXiv2022

Spaces of countable free set number and PFA

The main result of this paper is that, under PFA, for every {\em regular} space $X$ with $F(X) = ω$ we have $|X| \le w(X)^ω$; in particular, $w(X) \le \mathfrak{c}$ implies $|X| \le \mathfrak{c}$. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. We also show that regularity cannot be weakened to Hausdorff in this result because we can find in ZFC a Hausdorff space $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. In fact, this space $X$ has the {\em strongly anti-Urysohn} (SAU) property that any two infinite closed sets in $X$ intersect, which is much stronger than $F(X) = ω$. Moreover, any non-empty open set in $X$ also has size $2^\mathfrak{c}$, and thus answers one of the main problems of \cite{JShSSz} by providing in ZFC a SAU space with no isolated points.

math.GN↗

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.

math.GN↗

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.

math.GN↗

arXiv2020

On the cardinality of separable pseudoradial spaces

The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is, in ZFC, a regular (or just Hausdorff) SP of cardinality greater than $\mathfrak c$. While this question is left open, we establish a number of non-trivial results that we list: 1. It is consistent with Martin's Axiom and $\mathfrak c =\aleph_2$ that there is a countably tight and compact SP of cardinality $2^{\mathfrak c}$. 2. If $κ$ is a measurable cardinal then in the forcing extension obtained by adding $κ$ many Cohen reals, every countably tight regular SP space has cardinality at most $\mathfrak c$. 3. If $κ>\aleph_1$ Cohen reals are added to a model of GCH, then in the extension every pseudocompact SP space with a countable dense set of isolated points has cardinality at most $\mathfrak c$. 4. If $\mathfrak c\leq\aleph_2$, then there is a 0-dimensional SP space with a countable dense set of isolated points that has cardinal greater than $\mathfrak c$.

math.GN↗

arXiv2018

Densely k-separable compacta are densely separable

A space has $σ$-compact tightness if the closures of $σ$-compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable space is separable. The somewhat surprising answer is that this property, for compact spaces, implies that every dense set is separable. The path to this result relies on the known connections established between $π$-weight and the density of all dense subsets, or more precisely, the cardinal invariant $δ(X)$.

math.GN↗

arXiv2013

Regular spaces of small extent are omega-resolvable

We improve some results of Pavlov and of Filatova, respectively, concerning a problem of Malychin by showing that every regular space X that satisfies Delta(X)>ext(X) is omega-resolvable. Here Delta(X), the dispersion character of X, is the smallest size of a non-empty open set in X and ext(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are omega-resolvable. We also prove that any regular Lindelöf space X with |X|=Δ(X)=omega_1 is even omega_1-resolvable. The question if regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.

math.GN↗

arXiv2013

Strong colorings yield kappa-bounded spaces with discretely untouchable points

It is well-known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklossy and the first author, we show that this statement fails for countably compact regular spaces, and even for omega-bounded regular spaces. In fact, there are kappa-bounded counterexamples for every infinite cardinal kappa. The proof makes essential use of the so-called 'strong colorings' that were invented by the second author.

math.GN↗

arXiv2009

CH, a problem of Rolewicz and bidiscrete systems

We give a construction under $CH$ of a non-metrizable compact Hausdorff space $K$ such that any uncountable semi-biorthogonal sequence in $C(K)$ must be of a very specific kind. The space $K$ has many nice properties, such as being hereditarily separable, hereditarily Lindelöf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on $K$ are separable. However $K$ is not a Rosenthal compactum. We introduce the notion of bidiscrete systems in compact spaces and note that every infinite compact Hausdorff space $K$ must have a bidiscrete system of size $d(K)$, the density of $K$. This, in particular, implies that $C(K)$ has a biorthogonal system of size $d(K)$.

math.GN↗

arXiv2007

Projective $π$-character bounds the order of a $π$-base

All spaces below are Tychonov. We define the projective pi-character p(X) of a space X as the supremum of the values $πχ(Y)$ where Y ranges over all continuous images of X. Our main result says that every space X has a pi-base whose order is at most p(X), that is every point in X is contained in at most p(X)-many members of the pi-base. Since p(X) is at most t(X) for compact X, this provides a significant generalization of a celebrated result of Shapirovskii.

math.GN↗

arXiv2007

First countable spaces without point-countable $π$-base

We answer several questions of V. Tkačuk from [Point-countable $π$-bases in first countable and similar spaces, Fund. Math. 186 (2005), pp. 55--69.] by showing that (1) there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable $π$-base (in fact, the order of any $π$-base of the space is at least $\aleph_ω$); (2) if there is a $κ$-Suslin line then there is a first countable GO space of cardinality $κ^+$ in which the order of any $π$-base is at least $κ$; (3) it is consistent to have a first countable, hereditarily Lindel\" of regular space having uncountable $π$-weight and $ω_1$ as a caliber (of course, such a space cannot have a point-countable $π$-base).

math.GN↗

arXiv2007

Lindelof spaces of singular density

A cardinal lambda is called omega-inaccessible if for all mu < lambda we have mu^omega<lambda. We show that for every omega-inaccessible cardinal lambda there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindelof regular space of density lambda. This extends an analogous earlier result of ours that only worked for regular lambda.

math.LO↗

arXiv2007

Resolvability vs. almost resolvability

A space X is kappa-resolvable (resp. almost kappa-resolvable) if it contains kappa dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X). Answering a problem raised by Juhasz, Soukup, and Szentmiklossy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal {kappa} there is an almost 2^{kappa}-resolvable but not {omega}_1-resolvable space of dispersion character {kappa} .

math.GN↗

arXiv2006

D-forced spaces: a new approach to resolvability

We introduce a ZFC method that enables us to build spaces (in fact special dense subspaces of certain Cantor cubes) in which we have "full control" over all dense subsets. Using this method we are able to construct, in ZFC, for each uncountable regular cardinal $λ$ a 0-dimensional $T_2$, hence Tychonov, space which is $μ$-resolvable for all $μ< λ$ but not $λ$-resolvable. This yields the final (negative) solution of a celebrated problem of Ceder and Pearson raised in 1967: Are $ω$-resolvable spaces maximally resolvable? This method enables us to solve several other open problems concerning resolvability as well.

math.GN↗

arXiv2006

Resolvability of spaces having small spread or extent

In a recent paper O. Pavlov proved the following two interesting resolvability results: (1) If a space $X$ satisfies $Δ(X) > \ps(X)$ then $X$ is maximally resolvable. (2) If a $T_3$-space $X$ satisfies $Δ(X) > \pe(X)$ then $X$ is $ω$-resolvable. Here $\ps(X)$ ($\pe(X)$) denotes the smallest successor cardinal such that $X$ has no discrete (closed discrete) subset of that size and $Δ(X)$ is the smallest cardinality of a non-empty open set in $X$. In this note we improve (1) by showing that $Δ(X) >$ $\ps(X)$ can be relaxed to $Δ(X) \ge$ $\ps(X)$. In particular, if $X$ is a space of countable spread with $Δ(X) > ω$ then $X$ is maximally resolvable. The question if an analogous improvement of (2) is valid remains open, but we present a proof of (2) that is simpler than Pavlov's.

math.GN↗

arXiv2006

Resolvability and monotone normality

A space $X$ is said to be $κ$-resolvable (resp. almost $κ$-resolvable) if it contains $κ$ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets). $X$ is maximally resolvable iff it is $Δ(X)$-resolvable, where $Δ(X) = \min\{|G| : G \ne \emptyset {open}\}.$ We show that every crowded monotonically normal (in short: MN) space is $ω$-resolvable and almost $μ$-resolvable, where $μ= \min\{2^ω, ω_2 \}$. On the other hand, if $κ$ is a measurable cardinal then there is a MN space $X$ with $Δ(X) = κ$ such that no subspace of $X$ is $ω_1$-resolvable. Any MN space of cardinality $< \aleph_ω$ is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space $X$ with $|X| = Δ(X) = \aleph_ω$ such that no subspace of $X$ is $ω_2$-resolvable.

math.GN↗

arXiv2002

Characterizing continuity by preserving compactness and connectedness

Let us call a function $f$ from a space $X$ into a space $Y$ preserving if the image of every compact subspace of $X$ is compact in $Y$ and the image of every connected subspace of $X$ is connected in $Y$. By elementary theorems a continuous function is always preserving. Evelyn R. McMillan proved in 1970 that if $X$ is Hausdorff, locally connected and Frechet, $Y$ is Hausdorff, then the converse is also true: any preserving function $f:X\to Y$ is continuous. The main result of this paper is that if $X$ is any product of connected linearly ordered spaces (e.g. if $X = R^κ$) and $f:X \to Y$ is a preserving function into a regular space $Y$, then $f$ is continuous.

math.GN↗

arXiv1998

Strongly almost disjoint families, II

The relations M(kappa,lambda,mu)->B [resp. B(sigma)] meaning that if A subset [kappa]^lambda with |A|=kappa is mu-almost disjoint then A has property B [resp. has a sigma-transversal] had been introduced and studied under GCH by Erdos and Hajnal in 1961. Our two main results here say the following: Assume GCH and rho be any regular cardinal with a supercompact [resp. 2-huge] cardinal above rho. Then there is a rho-closed forcing P such that, in V^P, we have both GCH and M(rho^{(+rho+1)},rho^+,rho) not-> B [resp. M(rho^{(+rho+1)},lambda,rho) not-> B(rho^+) for all lambda =< rho^{(+rho+1)}].

math.LO↗