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I. Juhász

Publications and source records attributed to I. Juhász.

8 recordsLinked to original sources
arXiv2025

Some new results on $Δ$-spaces

A topological space $X$ is a $Δ$-space (or $X \in Δ$) if for any decreasing sequence $\{A_n : n < ω\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n : n < ω\}$ of open sets with empty intersection such that $A_n \subset U_n$ for all $n < ω$. In this note we prove the following results concerning $Δ$-spaces. 1) Every $T_3$ countably compact $Δ$-space is compact. 2) If there is a $T_1$ crowded Baire $Δ$-space then there is an inner model with a measurable cardinal. 3) If $X \in Δ$ and $cf \big(o(X) \big) > ω$ then $|X| < o(X)$. (Here $o(X)$ is the number of open subsets of $X$.) The first two of these provide full and/or partial solutions to problems raised in the literature, while the third improves a known result.

math.GN↗
arXiv2018

On maps preserving connectedness and /or compactness

We call a function $f: X\to Y$ $P$-preserving if, for every subspace $A \subset X$ with property $P$, its image $f(A)$ also has property $P$. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions is such a map continuous, has a long history. Our main result is that any non-trivial product function, i.e. one having at least two non-constant factors, that has connected domain, $T_1$ range, and is connectedness-preserving must actually be continuous. The analogous statement badly fails if we replace in it the occurrences of "connected" by "compact". We also present, however, several interesting results and examples concerning maps that are compactness-preserving and/or continuum-preserving.

math.GN↗
arXiv2016

Weakly linearly Lindelöf monotonically normal spaces are Lindelöf

We call a space $X$ {\it weakly linearly Lindelöf} if for any family $\mathcal{U}$ of non-empty open subsets of $X$ of regular uncountable cardinality $κ$, there exists a point $x\in X$ such that every neighborhood of $x$ meets $κ$-many elements of $\mathcal{U}$. We also introduce the concept of {\it almost discretely Lindelöf} spaces as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf. The main result of the paper is formulated in the title. It implies, among other things, that every weakly Lindelöf monotonically normal space is Lindelöf; this result seems to be new even for linearly ordered topological spaces. We show that, under the hypothesis $2^ω< ω_ω$, if the co-diagonal $Δ^c_X=(X\times X)\setminus Δ_X$ of a space $X$ is discretely Lindelöf, then $X$ is Lindelöf and has a weaker second countable topology; here $Δ_X=\{(x,x): x\in X\}$ is the diagonal of the space $X$. Moreover, the discrete Lindelöfness of $Δ^c_X$ together with the Lindelöf $Σ$-property of $X$ imply that $X$ has a countable network.

math.GN↗
arXiv1996

Combinatorial principles from adding Cohen reals

In this paper we first formulate several ``combinatorial principles'' concerning kappa \times omega matrices of subsets of omega and prove that they are valid in the generic extension obtained by adding any number of Cohen reals to any ground model V, provided that the parameter kappa is an omega-inaccessible regular cardinal in V. Then we present a large number of applications of these principles, mainly to topology. Some of these consequences had been established earlier in generic extensions obtained by adding omega_2 Cohen reals to ground models satisfying CH, mostly for the case kappa=omega_2.

math.LO↗
arXiv1996

Intersection properties of open sets, II.

A topological space is called P_2 ( P_3, P_{<omega} ) if and only if it does not contain two (three, finitely many) uncountable open sets with empty intersection. We show that (i) there are 0-dimensional P_{<omega} spaces of size 2^omega, (ii) there are compact P_{<omega} spaces of size omega_1, (iii) the existence of a Psi-like examples for a compact P_{<omega} space of size omega_1 is independent of ZFC, (iv) it is consistent that 2^omega is as large as you wish but every first countable (and so every compact) P_2 space has cardinality<=omega_1.

math.LO↗
arXiv1995

Forcing countable networks for spaces satisfying R(X^omega)=omega.

We show that all finite powers of a Hausdorff space X do not contain uncountable weakly separated subspaces iff there is a c.c.c poset P such that 1_P forces that ``X is a countable union of 0-dimensional subspaces of countable weight.'' We also show that this theorem is sharp in two different senses: (i) we can't get rid of using generic extensions, (ii) we have to consider all finite powers of $X$.

math.LO↗
arXiv1994

What makes a space have large weight?

We formulate several conditions (two of them are necessary and sufficient) which imply that a space of small character has large weight. In section 3 we construct a ZFC example of a first countable 0-dimensional space X of size 2^omega with w(X)=2^omega and nw(X)=omega, we show that CH implies the existence of a 0-dimensional space Y of size omega_1 with w(Y)=nw(Y)=omega_1 and chi(Y)=R(Y)=omega, and we prove that it is consistent that 2^omega is as large as you wish and there is a 0-dimensional space Z of size 2^omega such that w(Z)=nw(Z)=2^omega but chi(Z)=R(Z^omega)=omega.

math.LO↗