Tychonoff Expansions with Prescribed Resolvability Properties

TL;DR

This paper constructs Tychonoff space expansions with specific resolvability properties, demonstrating the existence of spaces with various combinations of resolvability and extraresolvability within ZFC, using the ${ m KID}$ expansion method.

Contribution

The authors introduce a method to produce Tychonoff expansions with prescribed resolvability properties, answering longstanding questions in topology.

Findings

Existence of expansions that are $ ext{omega}$-resolvable but not maximally resolvable.

Construction of spaces that are maximally resolvable but not extraresolvable.

Spaces that are both maximally resolvable and extraresolvable, but not strongly extraresolvable.

Abstract

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) and to Comfort and Garc\'ia-Ferreira (2001): (1) Is every $\omega$-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of ${\mathcal{KID}}$ expansion, the authors show that {\it every} suitably restricted Tychonoff topological space $(X,\sT)$ admits a larger Tychonoff topology (that is, an "expansion") witnessing such failure. Specifically the authors show in ZFC that if $(X,\sT)$ is a maximally resolvable Tychonoff space with $S(X,\sT)\leq\Delta(X,\sT)=\kappa$, then $(X,\sT)$ has Tychonoff expansions $\sU=\sU_i$ ($1\leq i\leq5$), with $\Delta(X,\sU_i)=\Delta(X,\sT)$ and $S(X,\sU_i)\leq\Delta(X,\sU_i)$, such that $(X,\sU_i)$ is: ($i=1$) $\omega$-resolvable but not maximally resolvable; ($i=2$) [if $\kappa'$ is regular, with $S(X,\sT)\leq\kappa'\leq\kappa$] $\tau$-resolvable for all $\tau<\kappa'$, but not $\kappa'$-resolvable; ($i=3$) maximally resolvable, but not extraresolvable; ($i=4$) extraresolvable, but not maximally resolvable; ($i=5$) maximally resolvable and extraresolvable, but not strongly extraresolvable.