On the descriptive complexity of homogeneous continua

TL;DR

This paper investigates the descriptive complexity of homogeneous continua within hyperspaces of subcontinua, revealing their analytic nature and embedding properties related to the space c_0.

Contribution

It demonstrates that the family of all homogeneous continua forms an analytic subspace containing a closed copy of c_0, highlighting their topological complexity.

Findings

The family of homogeneous continua is an analytic subspace.

Contains a topological copy of c_0 as a closed subset.

Applicable to hyperspaces of Euclidean and Hilbert cubes.

Abstract

It is shown that the family of all homogeneous continua in the hyperspace of all subcontinua of any finite-dimensional Euclidean cube or the Hilbert cube is an analytic subspace of the hyperspace which contains a topological copy of the linear space $c_0=\{(x_k)\in \mathbb R^\omega: \lim x_k=0\}$ as a closed subset.