Dominating and pinning down pairs for topological spaces

TL;DR

This paper introduces generalized concepts of dominating and pinning down pairs for topological spaces, extending previous notions, and investigates how these relate to the space's density under various conditions.

Contribution

It generalizes existing concepts of domination and pinning down pairs, providing answers to open problems about their implications for the density of topological spaces.

Findings

Established conditions under which certain pairs imply bounds on the density

Extended the framework of domination and pinning down pairs beyond previous special cases

Solved multiple open problems from prior research papers

Abstract

We call a pair of infinite cardinals $(\kappa,\lambda)$ with $\kappa > \lambda$ a dominating (resp. pinning down) pair for a topological space $X$ if for every subset $A$ of $X$ (resp. family $\mathcal{U}$ of non-empty open sets in $X$) of cardinality $\le \kappa$ there is $B \subset X$ of cardinality $\le \lambda$ such that $A \subset \overline{B}$ (resp. $B \cap U \ne \emptyset$ for each $U \in \mathcal{U}$). Clearly, a dominating pair is also a pinning down pair for $X$. Our definitions generalize the concepts introduced in [GTW] resp. [BT] which focused on pairs of the form $(2^\lambda,\lambda)$. The main aim of this paper is to answer a large number of the numerous problems from [GTW] and [BT] that asked if certain conditions on a space $X$ together with the assumption that $(2^\lambda,\lambda)$ or $((2^\lambda)^+,\lambda)$ is a pinning down pair or \dominating pair for $X$ would imply $d(X) \le \lambda$. [BT] A. Bella, V.V. Tkachuk, Exponential density vs exponential domination, preprint [GTW] G. Gruenhage, V.V. Tkachuk, R.G. Wilson, Domination by small sets versus density, Topology and its Applications 282 (2020)