Criteria for homogeneity of the Vietoris hyperspace Exp(X)

Determine the necessary and sufficient conditions on a topological space X under which the Vietoris hyperspace Exp(X) of all non-empty closed subsets of X is homogeneous.

Background

The paper studies homogeneity properties of hyperspaces, focusing on spaces derived from the double arrow A and the Sorgenfrey line S. The Vietoris hyperspace Exp(X) is the space of all non-empty closed subsets of X endowed with the Vietoris topology.

Known results indicate nuanced behavior: Exp([0,1]) is homeomorphic to the Hilbert cube (hence homogeneous), while for certain large cardinals κ, Exp(2^κ) is not homogeneous. This contextualizes the general problem of identifying conditions on X that guarantee homogeneity of Exp(X).