publication . Preprint . Article . 2015

Pinning Down versus Density

István Juhász; Lajos Soukup; Zoltán Szentmiklóssy;
Open Access English
  • Published: 31 May 2015
Abstract
The pinning down number $ {pd}(X)$ of a topological space $X$ is the smallest cardinal $\kappa$ such that for any neighborhood assignment $U:X\to \tau_X$ there is a set $A\in [X]^\kappa$ with $A\cap U(x)\ne\emptyset$ for all $x\in X$. Clearly, c$(X) \le {pd}(X) \le {d}(X)$. Here we prove that the following statements are equivalent: (1) $2^\kappa<\kappa^{+\omega}$ for each cardinal $\kappa$; (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 $\Delta(X)$ of a space $X$ is the smallest cardinality of a non-empty open subset of ...
Subjects
free text keywords: Mathematics - General Topology, Mathematics - Logic, 03E04, 03E10, 03E35, 54A25, 54A35}, Topological space, Combinatorics, Mathematical analysis, Mathematics

[1] U. Abraham, M. Magidor, Cardinal arithmetic, Handbook of Set Theory. Vols. 1, 2, 3, 1149-1227, Springer, Dordrecht, 2010.

[2] Taras Banakh, Alex Ravsky, Verbal covering properties of topological spaces, arXiv:1503.04480

[3] Saharon Shelah, Cardinal arithmetic for skeptics, American Math Soc Bulletin. New Series 26 (1992) 197-210. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences E-mail address: juhasz@renyi.hu Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences E-mail address: soukup@renyi.hu [OpenAIRE]

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publication . Preprint . Article . 2015

Pinning Down versus Density

István Juhász; Lajos Soukup; Zoltán Szentmiklóssy;