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Lajos Soukup

Publications and source records attributed to Lajos Soukup.

At least 73 records · Page 4Linked to original sources
arXiv2010

Dense families of countable sets below $c$

We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $κ$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both $T$ and $A$ have cardinality $κ$, (2) $|\bar{a}\cap T|=κ$ for each $a\in A$, (3) for each uncountable subset of $T$ contains some elements of $A$, and so (i) there is an almost disjoint family of subsets of the reals with size and chromatic number $κ$, (ii) there is a locally compact, locally countable $T_2$ space with cardinality spectrum $\{ω,κ\}$.

math.LO↗
arXiv2010

More on cardinal invariants of analytic P-ideals

Given an ideal $I$ on $ω$ let $a(I) $ ($\bar{a}(I)$) be minimum of the cardinalities of infinite (uncountable) maximal $I$-almost disjoint subsets of $[ω]^ω$, and denote $b_I$ and$d_I$ the unbounding and dominating numbers of $(ω^ω,\le_I)$. We show that (1) $a(I)>omega$ if $I$ is a summable ideal; (2) $a(Z)=ω$ and $\bar{a}(Z)\le a$ if $Z$ is a tall density ideal, (3) $b\le \bar{a}(I)$, and $b_I=b$ and $d_I=d$, for any analytic P-ideal $I$ on $ω$. Given an analytic $P$-ideal $I$ we investigate the relationship between the Sack, the $I$-bounding, $I$-dominating and $ω^ω$-bounding properties of a given poset $P$. For example, for the density zero ideal $Z$ we can prove: (i) a poset $P$ is $Z$-bounding iff it has the Sacks property, (ii) if $P$ adds a slalom capturing all ground model reals then $P$ is $Z$-dominating.

math.LO↗
arXiv2009

On splitting infinite-fold covers

Let $X$ be a set, $\ka$ be a cardinal number and let $\iH$ be a family of subsets of $X$ which covers each $x\in X$ at least $\ka$ times. What assumptions can ensure that $\iH$ can be decomposed into $κ$ many disjoint subcovers? We examine this problem under various assumptions on the set $X$ and on the cover $\iH$: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of $\real^{n}$ by polyhedra and by arbitrary convex sets. We focus on these problems mainly for infinite $κ$. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.

math.CO↗
arXiv2008

Indestructible colourings and rainbow Ramsey theorems

We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{ω_2}]^2\to 2$ establishing $ω_2 \not\to [(ω_1;ω)]^2_2$ such that some colouring $g:[ω_1]^2\to 2$ can not be embedded into $c$. It is also consistent that $2^{ω_1}$ is arbitrarily large, and a function $g$ establishes $2^{ω_1} \not\to [(ω_1,ω_2)]^2_{ω_1}$ such that there is no uncountable $g$-rainbow subset of $2^{ω_1}$. We also show that for each $k\in ω$ it is consistent with Martin's Axiom that the negative partition relation $ω_1 \not\to^* [(ω_1;ω_1)]_{k-bdd}$ holds.

math.LO↗
arXiv2007

Quasi-kernels and quasi-sinks in infinite graphs

Given a directed graph G=(V,E) an independent set A of the vertices V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A). Every finite directed graph has a quasi-kernel. The plain generalization for infinite graphs fails, even for tournaments. We investigate the following conjecture here: for any digraph G=(V,E) there is a a partition (V_0,V_1) of the vertex set such that the induced subgraph G[V_0] has a quasi-kernel and the induced subgraph G[V_1] has a quasi-sink.

math.CO↗
arXiv2007

Cardinal sequences of LCS spaces under GCH

We give full characterization of the sequences of regular cardinals that may arise as cardinal sequences of locally compact scattered spaces under GCH. The proofs are based on constructions of universal locally compact scattered spaces.

math.LO↗
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

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

Nagata's conjecture and countably compactifications in generic extensions

Nagata conjectured that every $M$-space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. This conjecture was refuted by Burke and van Douwen, and A. Kato, independently. However, we can show that there is a c.c.c. poset $P$ of size $2^ω$ such that in $V^P$ Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space $X\in V$ is an $M$-space in $V^P$ then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in $V^P$). In fact, we show that every first countable regular space from the ground model has a first countable countably compact extension in $V^P$, and then apply some results of Morita. As a corollary, we obtain that every first countable regular space from the ground model has a maximal first countable extension in model $V^P$.

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↗
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↗
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↗
arXiv2001

A tall space with small bottom

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if kappa^{<kappa}=kappa then there is such a space of height kappa^+ with only kappa many isolated points. This implies that there is a locally compact scattered space of height omega_2 with omega_1 isolated points in ZFC, solving an old problem of the first author.

math.LO↗
arXiv1999

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_{omega+1},aleph_omega)-->(aleph_1,aleph_0), it is consistent with GCH that the assertion in (c) does not hold and also that adding aleph_omega Cohen reals destroys the weak Freese-Nation property of (P(omega),subseteq)

math.LO↗
arXiv1998

Sticks and clubs

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side-by-side product of partial orders which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with not CH and Martin's Axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of omega_1 together with not CH and Martin's Axiom for countable p.o.-sets.

math.LO↗