Papers by István Juhász
Canadian mathematical bulletin, Feb 6, 2024
If X is a topological space and Y is any set, then we call a family F of maps from X to Y nowhere... more If X is a topological space and Y is any set, then we call a family F of maps from X to Y nowhere constant if for every non-empty open set U in X there is , f is not constant on U. We prove the following result that improves several earlier results in the literature. If X is a topological space for which C(X), the family of all continuous maps of X to R, is nowhere constant and X has a π-base consisting of connected sets then X is c-resolvable.
arXiv (Cornell University), Feb 8, 2017
Let us denote by Φ(λ, µ) the statement that B(λ) = D(λ) ω , i.e. the Baire space of weight λ, has... more Let us denote by Φ(λ, µ) the statement that 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 C in B(λ) picks up all the µ colors. We call a space X π-regular if it is Hausdorff and for every nonempty open set U in X there is a non-empty open set V such that V ⊂ U . We recall that a space X is called 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 < c(X) then X is µ-resolvable, i.e. contains µ pairwise disjoint dense subsets. (Here 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 [6], resp. Ortiz-Castillo and Tomita .
Israel Journal of Mathematics, Dec 27, 2022
It is an interesting, maybe surprising, fact that different dense subspaces of even "nice" topolo... more 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 from [1] 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 ≤ 2 2 min S ; (2) S = dd(X) holds for a regular space X iff S is ω-closed and sup S ≤ 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 P(ω)/f in and S is any set of uncountable regular cardinals < κ with |S| < min S, then there is a compactum C such that {ω, κ} ∪ S ⊂ dd(C), moreover λ / ∈ dd(C) whenever |S| + ω < cf (λ) < κ and cf (λ) / ∈ S. (ii) It is consistent to have a separable compactum C such that dd(C) is not ω 1 -closed.
arXiv (Cornell University), May 7, 2017
The pinning down number pd(X) of a topological space X is the smallest cardinal κ such that for e... more The pinning down number pd(X) of a topological space X is the smallest cardinal κ such that for every neighborhood assignment U on X there is a set of size κ that meets every member of U . Clearly, pd(X) ≤ d(X) and we call X a pd-example if pd(X) < d(X). We denote by S the class of all singular cardinals that are not strong limit. It was proved in [6] that TFAE: (1) S = ∅; (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 S = ∅ 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, S = ∅ in itself is not sufficient to imply the existence of a connected T 3.5 pd-example. But if there is µ ∈ S with µ ≥ 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 S \ c = ∅ even implies that there is a locally convex topological vector space pd-example.
arXiv (Cornell University), Oct 11, 2018
A space has σ-compact tightness if the closures of σ-compact subsets determines the topology. We ... more 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 kseparable 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 πweight and the density of all dense subsets, or more precisely, the cardinal invariant δ(X).
Cambridge University Press eBooks, Mar 21, 2017
We first formulate several "combinatorial principles" concerning κ × ω matrices of subsets of ω a... more We first formulate several "combinatorial principles" concerning κ × ω matrices of subsets of ω 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 κ is an ω-inaccessible regular cardinal in V . Then in section 4 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 ω 2 Cohen reals to ground models satisfying CH, mostly for the case κ = ω 2 .
Commentationes Mathematicae Universitatis Carolinae, 2005
We prove that (A) if a countably compact space is the union of countably many D subspaces then it... more We prove that (A) if a countably compact space is the union of countably many D subspaces then it is compact; (B) if a compact T 2 space is the union of fewer than N (R) = cov(M) left-separated subspaces then it is scattered. Both (A) and (B) improve results of Tkačenko from 1979; (A) also answers a question that was raised by Arhangel'ski ǐ and improves a result of Gruenhage.
Fundamenta Mathematicae, 2007
We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that (1) there is a... more We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] 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 minimum order of a π-base of the space can be made arbitrarily large); (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 Lindelöf regular space having uncountable π-weight and ω 1 as a caliber (of course, such a space cannot have a point-countable π-base).
arXiv (Cornell University), Feb 13, 2017
We consider 9 natural tightness conditions for topological spaces that are all variations on coun... more We consider 9 natural tightness conditions for topological spaces that are all variations on countable tightnes and investigate the interrelationships between them. Several natural open problems are raised.
arXiv (Cornell University), May 7, 1995
Improving a result of M. Rabus we force a normal, locally compact, 0-dimensional, Frechet-Uryson,... more Improving a result of M. Rabus we force a normal, locally compact, 0-dimensional, Frechet-Uryson, initially ω 1 -compact and non-compact space X of size ω 2 having the following property: for every open (or closed) set A in X we have |A| ≤ ω 1 or |X \ A| ≤ ω 1 .
Fundamenta Mathematicae, 2015
We improve some results of Pavlov and of Filatova, respectively, concerning a problem of Malychin... more We improve some results of Pavlov and of Filatova, respectively, concerning a problem of Malychin by showing that every regular space X that satisfies ∆(X) > e(X) is ω-resolvable. Here ∆(X), the dispersion character of X, is the smallest size of a non-empty open set in X and e(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 ω-resolvable. We also prove that any regular Lindelöf space X with |X| = ∆(X) = ω 1 is even ω 1 -resolvable. The question if regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.
Proceedings of the American Mathematical Society, 1992
We prove that if k is an uncountable regular cardinal and a compact T2 space X contains a free se... more We prove that if k is an uncountable regular cardinal and a compact T2 space X contains a free sequence of length k , then X also contains such a sequence that is convergent. This implies that under CH every nonfirst countable compact T2 space contains a convergent oi\-sequence and every compact T2 space with a small diagonal is metrizable, thus answering old questions raised by the first author and M. Husek, respectively.
arXiv (Cornell University), Feb 1, 2022
The main result of this paper is that, under PFA, for every regular space X with F (X) = ω we hav... more The main result of this paper is that, under PFA, for every regular space X with F (X) = ω we have |X| ≤ w(X) ω ; in particular, w(X) ≤ c implies |X| ≤ c. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X with F (X) = ω such that w(X) = c and |X| = 2 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) = c and |X| = 2 c . In fact, this space X has the 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 c , and thus answers one of the main problems of [8] by providing in ZFC a SAU space with no isolated points.
Canadian Journal of Mathematics, Oct 1, 1976
Introduction. Our method using CH is a blend of two earlier constructions and Ostaszewski ) of he... more Introduction. Our method using CH is a blend of two earlier constructions and Ostaszewski ) of hereditarily separable {HS), regular, non-Lindelôf, first countable spaces. [4] produces a much better space than ours in § 1 ; it has all of our properties except that it is not realcompact (which is probably more interesting), and it is countably compact as well; however, the construction works only under O, which implies the continuum hypothesis (CH) but is not equivalent to it. The argument of [2], like ours, just needs CH, but it is much more complicated, and it is not immediate that the space produced is locally compact or perfectly normal (although, in fact, it is; see the remark at the end of § 1). In § 2, we use a more complicated version of the technique in § 1 to construct a first countable, cardinality coi, HS, Dowker space. A Dowker space is a normal, Hausdorff space which is not countably paracompact. There is a known "real" Dowker space but all of its cardinal functions are large . There is a known HS Dowker space but its construction depends on the existence of a Souslin line . It was an old conjecture that the existence of a small cardinality (or small cardinal function) Dowker space depended on the existence of a Souslin line, and this conjecture is disproved by our construction. Using our technique and O (which implies both CH and the existence of a Souslin line) we can construct a first countable, cardinality coi, HS, Dowker space which is also locally compact and c-countably compact; but we choose the weaker hypothesis over the stronger conclusion. In § 2 we use Lusin sets in our construction. A subset L of the line is Lusin if L is uncountable and every nowhere dense subset of L is countable. If we assume CH, then there are Lusin sets in the line. However if we assume Martin's axiom and the negation of CH, then there are no Lusin sets in the line. If we assume Martin's axiom and the negation of CH, then there is no non-Lindelôf, first countable, regular topology on a subset of the line which refines the usual topology and has the property that the closure of a set in the two topologies differs by an at most countable set. Since our construction in § 1 yields just such a topology, both constructions are independent of the usual axioms for set theory. 1. The basic idea for obtaining this space is to start with the usual topology of the real numbers (R), which has many of the properties we want; in particular
Israel Journal of Mathematics, Sep 1, 2016
The pinning down number pd(X) of a topological space X is the smallest cardinal κ such that for a... more The pinning down number pd(X) of a topological space X is the smallest cardinal κ such that for any neighborhood assignment U : 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 • d(X) = pd(X) for any locally compact Hausdorff space X; • for every Hausdorff space X we have |X| ≤ 2 2 pd(X) and pd(X) < d(X) implies Δ(X) < 2 2 pd(X) ; • for every regular space X we have min{Δ(X), w(X)} ≤ 2 pd(X) and d(X) < 2 pd(X) , moreover pd(X) < d(X) implies Δ(X) < 2 pd(X) .
arXiv (Cornell University), Sep 4, 2006
A space X is said to be κ-resolvable (resp. almost κ-resolvable) if it contains κ dense sets that... more 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 ∆(X)-resolvable, where ∆(X) = min{|G| : G = ∅ 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 ∆(X) = κ such that no subspace of X is ω 1 -resolvable. Any MN space of cardinality < ℵ ω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = ∆(X) = ℵ ω such that no subspace of X is ω 2 -resolvable.
arXiv (Cornell University), Dec 2, 2002
In this paper we use a natural forcing to construct a left-separated topology on an arbitrary car... more In this paper we use a natural forcing to construct a left-separated topology on an arbitrary cardinal κ. The resulting left-separated space X κ is also 0-dimensional T 2 , hereditarily Lindelöf, and countably tight. Moreover if κ is regular then d(X κ ) = κ, hence κ is not a caliber of X κ , while all other uncountable regular cardinals are. This implies that some results of [A] and [JSz] are, consistently, sharp. We also prove it consistent that for every countable set A of uncountable regular cardinals there is a hereditarily Lindelöf T 3 space X such that ̺ = cf (̺) > ω is a caliber of X exactly if ̺ ∈ A.
arXiv (Cornell University), Sep 4, 2006
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Papers by István Juhász