A strengthening of the Cech-Pospisil theorem

摘要

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 X 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 λ but one can form λ many pairwise disjoint intersections of subfamilies of F.) The proof is based on a recent, still unpublished, lemma of G. Gruenhage.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 Cech-Pospisil 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.