Compactifiable classes of compacta (Q2324552)

In this paper, if \(\mathcal{C}\) and \(\mathcal{D}\) are classes of topological spaces, then it is said that \(\mathcal{C}\) and \(\mathcal{D}\) are equivalent, written \(\mathcal{C}\cong\mathcal{D}\), if each element of \(\mathcal{C}\) is homeomorphic to an element of \(\mathcal{D}\) and vice versa. The authors define a ``composition'' \(\mathcal{A}\) with composition space \(A\) and indexing space \(B\) to be a continuous map \(q:A\to B\). They define and refer to two special types of compositions: (i) a compact composition if \(A\) and \(B\) are metrizable compacta; (ii) a Polish composition if \(A\) and \(B\) are Polish spaces. Let us repeat the authors' Definition 2.4. Definition 2.4. A class \(\mathcal{C}\) of topological spaces is called compactifiable (resp. Polishable) if there is a compact (resp. Polish) composition of \(\mathcal{C}\), i.e., if there is a continuous map \(q:A\to B\) between metrizable compacta (resp. Polish spaces) such that \(\{q^{-1}(b)\mid b\in B\}\cong\mathcal{C}\). The paper consists of 5 sections, the first being an introduction. In Section 2, they obtain several conditions equivalent to compactifiability and Polishability (Theorems 2.10 and 2.11). Section 3 involves a study of connections between compactifiable or Polishable classes and hyperspaces. In Section 4, they study preservation of the properties under various constructions and obtain several examples. Section 5 is inspired by the construction of a universal arc-like continuum (see Theorem 12.22 of [\textit{S. B. Nadler jun.}, Continuum theory. An introduction. New York: Marcel (1992; Zbl 0757.54009)]). The authors modify this construction and prove in Corollary 5.8 that for every countable family \(\mathcal{P}\) of metrizable compacta, the class of \(\mathcal{P}\)-like spaces is compactifiable.