Tychonoff Expansions with Prescribed Resolvability Properties

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 García-Ferreira (2001): (1) Is every $\omega$-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of $\mathcal{K}\mathcal{I}\mathcal{D}$ 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 $S(X,\sT)\leqΔ(X,\sT)=κ$, then $(X,\sT)$ has Tychonoff expansions $\sU=\sU_i$ ($1\le i\le 5$), with $Δ(X,\sU_i)=Δ(X,\sT)$ and $S(X,\sU_i)\leqΔ(X,\sU_i)$, such that $(X,\sU_i)$ is: ($i=1$) $\omega$-resolvable but not maximally resolvable; ($i=2$) [if ${\kappa }^{\prime }$ is regular, with $S(X,\sT)\leqκ'\leqκ$] $\tau$-resolvable for all $\tau <{\kappa }^{\prime }$, but not ${\kappa }^{\prime }$-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.

25 pages, 0 figures