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 BX of cardinality ≤λ such that A B (resp. B U ot= for each U 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.