A strengthening of the Čech–Pospišil theorem

We prove the following result: If in a compact space X there is a λ-branching family of closed sets then X cannot be covered by fewer than λ many discrete subspaces. (A family of sets F is λ-branching iff | F | < λ but one can form λ many pairwise disjoint intersections of subfamilies of F .) The proof is based on a recent, still unpublished, lemma of G. Gruenhage. As a consequence, we obtain the following strengthening of the well-known Čech–Pospišil theorem: If X a is compact T 2 space such that all points x ∈ X have character χ ( x , X ) ⩾ κ then X cannot be covered by fewer than 2 κ many discrete subspaces.