Covering compacta by discrete and other separated sets

Juhasz and van Mill denote by dis(X) the least cardinal of a cover of X by discrete subspaces. They show that dis(X) ≥ c (and hence the answer to Question 1.1 is positive) for any compact crowded hereditarily normal X. In fact, this follows from their stronger result that for such X, rs(X) + ls(X) ≥ c, where rs(X) (resp., ls(X)) is the least cardinal of a cover of X by right(resp., left)-separated subspaces. Here we prove that the answer to Question 1.1 is positive, without any further assumptions. Indeed, this is a corollary to our more general result that the property of being the union of ≤ κ-many discrete subspaces (i.e., dis(X) ≤ κ) is preserved by perfect mappings, a result proven earlier for the case κ = ω by D. Burke and R. Hansell [1]. It is still not known if either ls(X) ≥ c or rs(X) ≥ c holds for any compact crowded X. In [4], it is noted that rs(X) is at least m, tbe least cardinal of a cover of the real line by meager sets, and Juhasz and Szentmiklossy [5] showed that ls(X) ≥ m also. We obtain the partial result that both rs(X) ≥ c and ls(X) ≥ c hold for first countable crowded compacta, provided c is a regular cardinal. In [4], it is noted that any counterexample to Question 1.1 contains a separable counterexample which must have cardinality c. Their argument clearly works for the right and left-separated questions too.