Resolvability and complete accumulation points

Abstract

We prove that:

I. For every regular Lindelöf 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 Lindelöf property can be weakened to the linear Lindelöf property.