Galois-Tukey connection involving sets of metrics (Q436067)

For a~metrizable space~\(X\) let \(\text{M}(X)\) denote the set of all metrics on~\(X\) compatible with the topology of~\(X\). For \(d_1,d_2\in \text{M}(X)\) we write \(d_1\preceq d_2\) if the identity mapping on~\(X\) as a~function from \((X,d_2)\) into \((X,d_1)\) is uniformly continuous. \textit{M.~Kada} proved in [``How many miles to \(\beta X\)? II. Approximations to \(\beta X\) versus cofinal types of sets of metrics'', Topology Appl. 157, No. 8, 1460--1464 (2010; Zbl 1195.54053)] that the relation~\(\preceq\) is Galois-Tukey equivalent to the relation~\(\leq^*\) of the eventual dominance of functions in~\(\omega^\omega\) provided that \(X\) is a locally compact metrizable space such that the first Cantor-Bendixson derivative~\(X^{(1)}\) of~\(X\) is noncompact. Let \(\text{PC}(X)\) denote the set of all pairs of disjoint closed sets of~\(X\) and let \(\text{Sep}\) be the binary relation between \(\text{PC}(X)\) and \(\text{M}(X)\) defined by \((A,B)\mathrel{\text{Sep}}d\) if \(d(A,B)>0\) for \((A,B)\in\text{PC}(X)\) and \(d\in \text{M}(X)\).NEWLINENEWLINEIn the paper under the review the authors show under the same hypotheses that also the relation \(\text{Sep}\) as a~triple \((\text{PC}(X),\text{M}(X),\text{Sep})\) is Galois-Tukey equivalent to~\(\leq^*\). For \((A,B)\in \text{PC}(X)\), \(d,d_1,d_2\in \text{M}(X)\), and \(\varepsilon>0\) write \((A,B)\mathrel{\text{Sep}}_\varepsilon d\) if \(d(A,B)\geq\varepsilon\), and \(d_1\preceq_\varepsilon d_2\) if for \(p,q\in X\), \(d_1(p,q)\geq\varepsilon\) implies \(d_2(p,q)\geq\varepsilon\). A~separable metrizable space~\(X\) can be regarded as a~subset of the Hilbert cube \(\mathbf H=[0,1]^\omega\) and then let \(X^*=\text{cl}_{\mathbf H}(X)\setminus X\) and let \(\mathcal{K}(X^*)\) denote the space of compact subsets of~\(X^*\). In an attempt to solve a~question of Todorčević about the existence of the Galois-Tukey equivalence \((\text{M}(X),\preceq)\equiv (\omega^\omega\times\mathcal{K}(X^*),{\leq^*}\times{\subseteq})\) provided that \(X\)~is a~separable metrizable space such that \(X^{(1)}\) is noncompact the authors prove that the relational structures \((\omega^\omega\times\mathcal{K}(X^*),{\leq}\times{\subseteq})\), \((\text{M}(X),\preceq_1)\), and \((\text{PC}(X),\text{M}(X),\text{Sep}_1)\) are Galois-Tukey equivalent where for \(f,g\in\omega^\omega\), \(f\leq g\) if \(f(n)\leq g(n)\) for all \(n\in\omega\).