Dominating and pinning down pairs for topological spaces

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)