Compactifiable classes of compacta

TL;DR

This paper introduces and studies the concepts of compactifiable and Polishable classes of metrizable compact spaces, exploring their characterizations, properties, and relations to hyperspaces.

Contribution

It defines the notions of compactifiable and Polishable classes, provides characterizations, and investigates their properties and preservation under various constructions.

Findings

Characterizations of compactifiable and Polishable classes

Relations to hyperspaces and their properties

Conditions for preservation under constructions

Abstract

We introduce the notion of compactifiable classes -- these are classes of metrizable compact spaces that can be up to homeomorphic copies ``disjointly combined'' into one metrizable compact space. This is witnessed by so-called compact composition of the class. Analogously, we consider Polishable classes and Polish compositions. The question of compactifiability or Polishability of a class is related to hyperspaces. Strongly compactifiable and strongly Polishable classes may be characterized by the existence of a corresponding family in the hyperspace of all metrizable compacta. We systematically study the introduced notions -- we give several characterizations, consider preservation under various constructions, and raise several questions.