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 U of non-empty open sets in X) of cardinality $\leq\kappa$ there is $B\subset X$ of cardinality $\leq\lambda$ such that $A \subset \overline B$ (resp. $B \cap U \not=\emptyset$ 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 [4] resp. [3] 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 [4] and [3] 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 $d(X)\leq\lambda$.
This paper is dedicated to the memory of Phil Zenor.