Homogeneity of the hyperspace C2(S) over the Sorgenfrey line

Determine whether the hyperspace C2(S), consisting of all unions of at most two non-empty closed intervals in the Sorgenfrey line S, is homogeneous.

Background

Section 3 shows C1(S) is homogeneous by establishing a homeomorphism between the space of single closed intervals and a suitable product-order space. This motivates investigating the next finite case.

The problem asks whether homogeneity extends from single intervals to unions of at most two closed intervals in S, a space with well-known non-metrizable and non-second-countable properties that often complicate homogeneity arguments.