There is no compact metrizable space containing all continua as unique components
General Topology
2020-01-31 v1
Abstract
We answer a question of Piotr Minc by proving that there is no compact metrizable space whose set of components contains a unique topological copy of every metrizable compactification of a ray (i.e. a half-open interval) with an arc (i.e. closed bounded interval) as the remainder. To this end we use the concept of Borel reductions coming from Invariant descriptive set theory. It follows as a corollary that there is no compact metrizable space such that every continuum is homeomorphic to exactly one component of this space.
Cite
@article{arxiv.2001.11281,
title = {There is no compact metrizable space containing all continua as unique components},
author = {Benjamin Vejnar},
journal= {arXiv preprint arXiv:2001.11281},
year = {2020}
}