The dimension of hyperspaces of non-metrizable continua

TL;DR

This paper investigates the properties of hyperspaces of non-metrizable continua, establishing conditions under which these hyperspaces are not C-spaces based on the dimension and indecomposability of the original continuum.

Contribution

It generalizes existing results from metric to non-metrizable continua, providing new insights into the dimensional and topological properties of their hyperspaces.

Findings

If dim X > 1, then C(X) is not a C-space.

If dim X = 1 and X is hereditarily indecomposable, then dim C(X) = 2 or C(X) is not a C-space.

Abstract

We prove that, for any Hausdorff continuum X, if dim X > 1 then the hyperspace C(X) of subcontinua of X is not a C-space; if dim X = 1 and X is hereditarily indecomposable then dim C(X) = 2 or C(X) is not a C-space. This generalizes results known for metric continua.