Exactly $n$-resolvable Topological Expansions

TL;DR

This paper investigates the existence and construction of topological expansions that are exactly n-resolvable, focusing on finite n, and explores conditions for quasi-regularity and property preservation.

Contribution

It proves that every n-resolvable space admits an exactly n-resolvable expansion, characterizes when such expansions are quasi-regular, and discusses property preservation in expansions.

Findings

Every n-resolvable space admits an exactly n-resolvable expansion.

No universal quasi-regular expansion exists for all n-resolvable spaces.

Conditions are identified under which expansions preserve quasi-regularity and other properties.

Abstract

For $\kappa$ a cardinal, a space $X=(X,\sT)$ is $\kappa$-{\it resolvable} if $X$ admits $\kappa$-many pairwise disjoint $\sT$-dense subsets; $(X,\sT)$ is {\it exactly} $\kappa$-{\it resolvable} if it is $\kappa$-resolvable but not $\kappa^+$-resolvable. The present paper complements and supplements the authors' earlier work, which showed for suitably restricted spaces $(X,\sT)$ and cardinals $\kappa\geq\lambda\geq\omega$ that $(X,\sT)$, if $\kappa$-resolvable, admits an expansion $\sU\supseteq\sT$, with $(X,\sU)$ Tychonoff if $(X,\sT)$ is Tychonoff, such that $(X,\sU)$ is $\mu$-resolvable for all $\mu<\lambda$ but is not $\lambda$-resolvable (cf. Theorem~3.3 of \cite{comfhu10}). Here the "finite case" is addressed. The authors show in ZFC for $1<n<\omega$: (a) every $n$-resolvable space $(X,\sT)$ admits an exactly $n$-resolvable expansion $\sU\supseteq\sT$; (b) in some cases, even with $(X,\sT)$ Tychonoff, no choice of $\sU$ is available such that $(X,\sU)$ is quasi-regular; (c) if $n$-resolvable, $(X,\sT)$ admits an exactly $n$-resolvable quasi-regular expansion $\sU$ if and only if either $(X,\sT)$ is itself exactly $n$-resolvable and quasi-regular or $(X,\sT)$ has a subspace which is either $n$-resolvable and nowhere dense or is $(2n)$-resolvable. In particular, every $\omega$-resolvable quasi-regular space admits an exactly $n$-resolvable quasi-regular expansion. Further, for many familiar topological properties $\PP$, one may choose $\sU$ so that $(X,\sU)\in\PP$ if $(X,\sT)\in\PP$.