Homogeneity of the hyperspace Exp(A) of the double arrow space

Determine whether the Vietoris hyperspace Exp(A) of the double arrow space A is homogeneous.

Background

The double arrow space A (Alexandroff double arrow) is a classic compact, non-metrizable, linearly ordered topological space. Arhangel’skii asked whether its Vietoris hyperspace Exp(A) is homogeneous.

This paper partially advances understanding by proving that certain related hyperspaces over A are not homogeneous (specifically, C_m(A) for all m ≥ 1), while other hyperspaces (such as Sc(A)) are homogeneous, bringing the resolution of this question closer.