Universal spaces for finite-dimensional closed images of locally compact metric spaces (Q1295302)

The authors provide spaces as in the title of the paper. In the separable case (and \(\dim\leq n\)) the universal space \(L(n)\) is obtained as a special quotient of \(\mu^n\smallsetminus\{*\}\) (Menger's universal compactum minus a point). In the non-separable case -- weight \(\kappa\) -- one takes a quotient of the sum of \(\kappa\) copies of \(L(n)\), employing a family \(\mathcal K\) of countable subsets of \(\kappa\). The universality of the resulting space is equivalent to the following properties of \(\mathcal K\): its cardinality should be \(\kappa\) and given any other family \(\mathcal C\) of countable subsets of \(\kappa\) of size \(\kappa\) there is a permutation \(\pi\) of \(\kappa\) such that for every \(C\in\mathcal C\) there is \(K\in\mathcal K\) such that \(\pi[C]\subseteq K\). If the GCH holds then every cardinal carries such a family but the authors leave open the question whether ZFC suffices for this.