Covering compacta by discrete and other separated sets

We show that if a space X is the union of not more than κ-many discrete subspaces, where κ is an infinite cardinal, then the same holds for any perfect image of X. It follows that a compact Hausdorff space with no isolated points can never be covered by fewer than continuum many discrete subspaces; this answers a question of I. Juhász and J. van Mill. We also consider coverings by right-separated and left-separated subspaces.