Galois-Tukey connection involving sets of metrics. (English) Zbl 1248.54015

Kada, Masaru; Yoshinobu, Yasuo

Tsukuba J. Math. 36, No. 1, 53-66 (2012).

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. 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 \(d\in \text{M}(X)\).
In 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 \(n\in\omega\).

Reviewer: Miroslav Repický (Košice)

MSC:

54E35	Metric spaces, metrizability
03E17	Cardinal characteristics of the continuum
06A07	Combinatorics of partially ordered sets
54D35	Extensions of spaces (compactifications, supercompactifications, completions, etc.)

Keywords:

Smirnov compactification; Stone-Čech compactification; metrizable space; generalized Galois-Tukey connection

Citations:

Zbl 1195.54053