Covering compacta by discrete subspaces

For any space X, denote by dis (X) the smallest (infinite) cardinal κ such that κ many discrete subspaces are needed to cover X. It is easy to see that if X is any crowded (i.e. dense-in-itself) compactum then dis (X) ≥ m, where m denotes the additivity of the meager ideal on the reals. It is a natural, and apparently quite difficult, question whether in this inequality m could be replaced by c. Here we show that this can be done if X is also hereditarily normal. Moreover, we prove the following mapping theorem that involves the cardinal function dis (X). If f : X → Y is a continuous surjection of a countably compact T