Covering compacta by discrete subspaces
Abstract 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 2 space X onto a perfect T 3 space Y then | { y ∈ Y : f −1 y  is countable } | ⩽ dis ( X ) .
A strengthening of the Čech–Pospišil theorem
Abstract 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 The proof is based on a recent, still unpublished, lemma of G. Gruenhage.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 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.
Cardinalities of dense sets
Abstract A cardinal invariant on a topological space X , called its strong density, is introduced as the supremum of the densities of the dense subsets of X . The relationships of the strong density to the density and the π-weight are examined both in general and in the particular classes of suborderable spaces and hyperspaces. An example is constructed of a Hausdorff space X with strong density m such that X 2 has strong density larger than m .