Discrete sets of singular cardinality

Abstract

Let κ \kappa be a singular cardinal. In Fleissner’s thesis, he showed that in normal spaces X X , certain discrete sets Y Y of cardinality κ \kappa (called here sparse) which are > κ > \kappa -separated are, in fact, separated. In Watson’s thesis, he proves the same for countably paracompact spaces X X . Here we improve these results by making no assumption on the space X X . As a corollary, we get that assuming V = L V = L , ℵ 1 {\aleph _1} ,-paralindelöf T 2 {T_2} , spaces of character ⩽ ω 2 \leqslant {\omega _2} , are collectionwise Hausdorff.