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 ) .
Covering by discrete and closed discrete sets
Abstract Say that a cardinal number κ is small relative to the space X if κ Δ ( X ) , where Δ ( X ) is the least cardinality of a non-empty open set in X . We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ -space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.
π(X) = δ(X) for compact X
Abstract We prove that if X is a compact T 2 space (and x ϵ X ) and π( X )=κ (πχ( x, X )=κ), then there is a dense subset Y ⊂ X (resp. a set Y ⊂ X with x ϵ Ȳ ) such that d ( Y )=κ (resp. x ∉ Z for any Z ⊂ Y with |Z| G D α ( X ) is always determined if X is compact T 2 .