Domination by small sets versus density

Abstract

Given an infinite cardinal κ, we prove that a regular space X must have density not exceeding κ if and only if every subset of X of power (2κ)+ is κ-dominated, i.e., contained in the closure of a set of cardinality κ. Thus, it is natural to study the spaces in which all subsets of cardinality not exceeding 2κ are κ-dominated; this property will be called exponential κ-domination. The spaces with exponential ω-domination first appeared in the paper [5] because they are weakly exponentially separable. We will prove that the class of spaces with exponential κ-domination is preserved by continuous images, open subspaces, Σ2κ-products and κ-unions. It is shown that the cardinal κ+ is a caliber of any space X with exponential κ-domination. A Čech-complete space features exponential κ-domination if and only if its density does not exceed κ. However, the density of countably compact spaces with exponential ω-domination has no upper bound. If a regular space X has a dense subspace Y with πχ(Y)≤2κ then d(X)≤κ if and only if X features exponential κ-domination. We conclude the paper with an example of a non-separable Tychonoff space of countable pseudocharacter with exponential ω-domination.