Regular spaces of small extent are $\omega $-resolvable
Volume 228 / 2015
Abstract
We improve some results of Pavlov and Filatova, concerning a problem of Malykhin, by showing that every regular space $X$ that satisfies $\varDelta (X)>\operatorname {\rm e}(X)$ is ${\omega }$-resolvable. Here $\varDelta (X)$, the dispersion character of $X$, is the smallest size of a non-empty open set in $X$, and $\operatorname {\rm e}(X)$, the extent of $X$, is the supremum of the sizes of all closed-and-discrete subsets of $X$. In particular, regular Lindelöf spaces of uncountable dispersion character are ${\omega }$-resolvable.
We also prove that any regular Lindelöf space $X$ with $|X|=\varDelta (X)=\omega _1$ is even ${\omega _1}$-resolvable. The question whether regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.
