Homogeneity of the hyperspace C(A) of finite unions of intervals in the double arrow space

Determine whether the hyperspace C(A), consisting of all finite unions of non-empty closed intervals in the double arrow space A, is homogeneous.

Background

The paper proves that for every m ≥ 1, the hyperspace Cm(A) of unions of at most m non-empty closed intervals in A is not homogeneous. This result suggests exploring the unbounded finite union case.

The posed question asks if allowing an arbitrary finite number of intervals (rather than bounding by m) alters the homogeneity status for A, which could reveal structural differences between bounded and unbounded finite hyperspaces.