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 T2 space X onto a perfect T3 space Y then |{y ∈ Y : f−1y is countable}| dis(X). © 2006 Elsevier B.V. All rights reserved.