Dominating and pinning down pairs for topological spaces - Juhasz, Istvan; Soukup, Lajos; Szentmiklóssy, Zoltan (Topology Proceedings, 2021)
We call a pair of infinite cardinals ($κ,λ$) with $κ > λ$ 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 $≤κ$ there is $B\subset X$ of cardinality $≤λ$ such that $A \subset \overline B$ (resp. $B \cap U ot=\emptyset$ 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 ($2^λ,λ$). 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 ($2^λ,λ$) or ($(2^λ)^+,λ$) is a pinning down pair or dominating pair for X would imply $d(X)≤λ$. This paper is dedicated to the memory of Phil Zenor.