Resolvability and complete accumulation points

A. E. Lipin

Published 2023-01-30Version 1

We prove that: I. For every regular Lindel\"of space $X$ if $|X|=\Delta(X)$ and $\mathrm{cf}|X|\ne\omega$, then $X$ is maximally resolvable; II. For every regular countably compact space $X$ if $|X|=\Delta(X)$ and $\mathrm{cf}|X|=\omega$, then $X$ is maximally resolvable. Here $\Delta(X)$, the dispersion character of $X$, is the minimum cardinality of a nonempty open subset of $X$. Statements I and II are corollaries of the main result: for every regular space $X$ if $|X|=\Delta(X)$ and every set $A\subseteq X$ of cardinality $\mathrm{cf}|X|$ has a complete accumulation point, then $X$ is maximally resolvable. Moreover, regularity here can be weakened to $\pi$-regularity, and the Lindel\"of property can be weakened to the linear Lindel\"of property.