Spaces of non-uniform ultrafilters in spaces of uniform ultrafilters
For an infinite cardinal \(\kappa\), let U(\(\kappa\)) denote the space of uniform ultrafilters on \(\kappa\), and denote \(NU(\kappa)=\beta \kappa \setminus U(\kappa)\). It is known that NU(\(\gamma\)) is not normal if \(\gamma\) is regular and not a strong limit [\textit{V. I. Malyhin}, Soviet Math., Doklady 14 (1973), 1112-1115 (1974), translation from Doklady Akad. Nauk SSSR 211, 781-783 (1973; Zbl 0296.54017)]. A point \(x\in X\) is a non-normality point of X if \(X\setminus \{x\}\) is not normal, and a butterfly point if in \(X\setminus \{x\}\) there are disjoint closed subsets, each having x in its X-closure. The authors address the question under what conditions every point of \(\kappa\) is a butterfly point c.q. a non-normality point. For this, they develop a condition, expressed as \(\rho (\kappa)=2^{\kappa}\), and they prove that if \(\kappa\) is regular, it is consistent with all consistent cardinal arithmetic that this condition holds. Their main result then reads: Theorem. If \(\rho (\kappa)=2^{\kappa}\), then (a) every point of U(\(\kappa\)) is a butterfly point; and (b) if also \(\sup \{2^{\lambda}:\lambda <cf(2^{\kappa})\}=2^{\kappa}\), then every point of U(\(\kappa\)) is a non-normality point. - This implies \textit{N. M. Warren}'s result [Proc. Am. Math. Soc. 33, 599-606 (1972; Zbl 0241.54016)] that under CH every point of U(\(\omega\)) is a non-normality point, but even for this particular case \(\kappa =\omega\) the present result is much stronger. More generally, under GCH all non-isolated points of all \(\beta\kappa\) are non-normality points. The proof rests on embedding the non-normal space NU(cf 2\({}^{\kappa})\), for each \(x\in U(\kappa)\), as a closed subspace in U(\(\kappa\))\(\setminus \{x\}\); hence the title of the paper.
scientific article; zbMATH DE number 4180292
scientific article