Pinning down versus density

Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán

doi:10.1007/s11856-016-1389-3

Cited by 5 publications

(7 citation statements)

References 4 publications

(6 reference statements)

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“…So it follows from Corollary 2.3 that for any space X with ∆(X) = |X| < ℵ ω we have d(X) = pd(X). This in turn, by Lemma 2.2 of [6], implies d(X) = pd(X) whenever |X| < ℵ ω , a result first proved in Theorem 5.1 of [2].…”

Section: Results For Pinning Down Pairsmentioning

confidence: 67%

Dominating and pinning down pairs for topological spaces

Juhász¹,

Soukup²,

Szentmiklóssy³

2020

Preprint

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We call a pair of infinite cardinals (κ, λ) with κ > λ a dominating (resp. pinning down) pair for a topological space X if for every subset A ofClearly, a dominating pair is also a pinning down pair for X. Our definitions generalize the concepts introduced in [4] resp. [3] which focused on pairs of the form (2 λ , λ).The main aim of this paper is to answer a large number of the numerous problems from [4] and [3] that asked if certain conditions on a space X together with the assumption that (2 λ , λ) or ((2 λ ) + , λ) is a pinning down pair or dominating pair for X would imply d(X) ≤ λ.

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Section: Results For Pinning Down Pairsmentioning

confidence: 67%

Dominating and pinning down pairs for topological spaces

Juhász¹,

Soukup²,

Szentmiklóssy³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…But [6,Theorem 3.3] implies that if there is µ ∈ S with µ ≥ c then there is a neat 0dimensional T 2 , hence T 3.5 pd-example of size ≥ c, completing the proof of the first sentence in Theorem 1.3.…”

Section: Proof Of Theorem 13: Connected Topological Group Pd-examplesmentioning

confidence: 87%

“…If µ ∈ S then there is a cardinal λ satisfying cf(µ) ≤ λ < µ and 2 λ > µ. The construction theorem [6,Theorem 3.3], in fact a simplified version of it, then yields a 0-dimensional T 2 space Y such that pd(Y ) ≤ λ < d(Y ) = µ and w(Y ) ≤ 2 λ .…”

Section: Proofmentioning

confidence: 99%

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Connected and/or topological group pd-examples

Juhász¹,

Mill²,

Soukup³

et al. 2017

Preprint

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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.

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“…Remark 5.7. Answering Problems 5.5, 5.6 Juhász, Soukup and Szentmiklóssy [13] proved the equivalence of the following statements:…”

mentioning

confidence: 95%

Verbal covering properties of topological spaces

Банах

Ravsky

2016

Topology and its Applications

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For any topological space $X$ we study the relation between the universal uniformity $\mathcal U_X$, the universal quasi-uniformity $q\mathcal U_X$ and the universal pre-uniformity $p\mathcal U_X$ on $X$. For a pre-uniformity $\mathcal U$ on a set $X$ and a word $v$ in the two-letter alphabet $\{+,-\}$ we define the verbal power $\mathcal U^v$ of $\mathcal U$ and study its boundedness numbers $\ell(\mathcal U^v)$ and $\bar \ell(\mathcal U^v)$. The boundedness numbers of the (Boolean operations over) the verbal powers of the canonical pre-uniformities $p\mathcal U_X$, $q\mathcal U_X$ and $\mathcal U_X$ yield new cardinal characteristics $\ell^v(X)$, $\bar \ell^v(X)$, $q\ell^v(X)$, $q\bar \ell^v(X)$, $u\ell(X)$ of a topological space $X$, which generalize all known cardinal topological invariants related to (star)-covering properties. We study the relation of the new cardinal invariants $\ell^v$, $\bar \ell^v$ to classical cardinal topological invariants such as Lindel\"of number $\ell$, density $d$, and spread $s$. The simplest new verbal cardinal invariant is the foredensity $\ell^-(X)$ defined for a topological space $X$ as the smallest cardinal $\kappa$ such that for any neighborhood assignment $(O_x)_{x\in X}$ there is a subset $A\subset X$ of cardinality $|A|\le\kappa$ that meets each neighborhood $O_x$, $x\in X$. It is clear that $\ell^-(X)\le d(X)\le \ell^-(X)\cdot \chi(X)$. We shall prove that $\ell^-(X)=d(X)$ if $|X|<\aleph_\omega$. On the other hand, for every singular cardinal $\kappa$ (with $\kappa\le 2^{2^{cf(\kappa)}}$) we construct a (totally disconnected) $T_1$-space $X$ such that $\ell^-(X)=cf(\kappa)<\kappa=|X|=d(X)$.Comment: 20 pages, many diagram

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Pinning down versus density

Cited by 5 publications

References 4 publications

Dominating and pinning down pairs for topological spaces

Dominating and pinning down pairs for topological spaces

Connected and/or topological group pd-examples

Verbal covering properties of topological spaces

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