Hyperspaces of the double arrow

Let A and S denote the double arrow of Alexandroff and the Sorgenfrey line, respectively. We show that for any n≥ 1, the space of all unions of at most n closed intervals of A is not homogeneous. We also prove that the spaces of non-trivial convergent sequences of A and S are homogeneous. This partially solves an open question of A. Arhangel'ski{i}. In contrast, we show that the space of closed intervals of S is homogeneous.