Discrete Sets of Singular Cardinality

Abstract

Let $\kappa$ be a singular cardinal. In Fleissner’s thesis, he showed that in normal spaces $X$, certain discrete sets $Y$ of cardinality $\kappa$ (called here sparse) which are $< \kappa$-separated are, in fact, separated. In Watson’s thesis, he proves the same for countably paracompact spaces $X$. Here we improve these results by making no assumption on the space $X$. As a corollary, we get that assuming $V = L$, ${\aleph _1}$,-paralindelöf ${T_2}$, spaces of character $\leqslant {\omega _2}$, are collectionwise Hausdorff.