Pinning Down versus Density

TL;DR

This paper investigates the relationships between the pinning down number, density, and dispersion character of topological spaces, establishing equivalences under set-theoretic assumptions and proving bounds for various classes of spaces.

Contribution

It provides new characterizations of when the pinning down number equals the density, answers open questions, and explores the set-theoretic consistency of certain inequalities.

Findings

${d}(X)={pd}(X)$ for all Hausdorff spaces under certain set-theoretic conditions

Locally compact Hausdorff spaces satisfy ${d}(X)={pd}(X)$

Bounds on the size and dispersion character of spaces with ${pd}(X)<{d}(X)$

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 $X$. We also show that if ${pd}(X)<{d}(X)$ then $X$ has an open subspace $Y$ with ${pd}(Y)<{d}(Y)$ and $|Y| = \Delta(Y)$, moreover the following three statements are equiconsistent: (i) There is a singular cardinal $\lambda$ with $pp(\lambda)>\lambda^+$, i.e. Shelah's Strong Hypothesis fails; (ii) there is a 0-dimensional Hausdorff space $X$ such that $|X|=\Delta(X)$ is a regular cardinal and ${pd}(X)<{d}(X)$; (iii) there is a topological space $X$ such that $|X|=\Delta(X)$ is a regular cardinal and ${pd}(X)<{d}(X)$. We also prove that $\bullet$ ${d}(X)={pd}(X)$ for any locally compact Hausdorff space $X$; $\bullet$ for every Hausdorff space $X$ we have $|X|\le 2^{2^{{pd}(X)}}$ and ${pd}(X)<{d}(X)$ implies $\Delta(X)< 2^{2^{{pd}(X)}}$; $\bullet$ for every regular space $X$ we have $\min\{\Delta(X),\, w(X)\}\le 2^{{pd}(X)}\,$ and ${d}(X)<2^{{pd}(X)},\,$ moreover ${pd}(X)<{d}(X)$ implies $\,\Delta(X)< {2^{{pd}(X)}}$.