Resolvability and complete accumulation points

TL;DR

This paper establishes conditions under which certain regular topological spaces are maximally resolvable, extending previous results by weakening regularity and Lindel"of assumptions.

Contribution

It proves that spaces with specific cardinality and accumulation point properties are maximally resolvable, generalizing prior theorems with weaker conditions.

Findings

Regular Lindel"of spaces with equal size and dispersion character are maximally resolvable.

Regular countably compact spaces with equal size and countable cofinality are maximally resolvable.

The main result links accumulation points to resolvability under weakened regularity and Lindel"of conditions.

Abstract

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.