Sigma-fragmentable spaces that are not countable unions of fragmentable subspaces

In this paper we show that although (ℓ∞/c0,weak) is sigma-fragmented by some metric it cannot be decomposed into countably many fragmentable subspaces. Likewise, we show that although the dual space of the continuous functions defined on the double arrow space is weak∗ sigma-fragmented by some metric, it too cannot be decomposed into a countable union of fragmentable subspaces.