Non-homeomorphic disjoint spaces whose union is \(\omega^*\)
Three decompositions are constructed of the form \(\omega^* = \bigcup\{C_ \alpha: \alpha < \kappa\}\) with \(C_ \alpha\) dense, countably compact, pairwise disjoint, pairwise non-homeomorphic and of cardinality prescribed in advance. In some cases the sets \(C_ \alpha\) may be chosen to satisfy certain additional conditions. In particular one can require that cardinality of \(C_ \alpha\) equals continuum and there is no one-to-one continuous map from \(C_ \alpha\) into \(\omega^* - C_ \alpha\) for any \(\alpha < \kappa \leq 2^ \mathbb{C}\). The methods used for the constructions depend on so called \(p\)-closures, disjoint refinements, and the existence of \(2^ \mathbb{C}\)-many minimal types in the sense of Rudin-Frolík. The results presented here are strenthenings of those of \textit{W. Comfort} and \textit{C. Waiveris} [Usp. Mat. Nauk 35, No. 3(213), 67-77 (1980; Zbl 0435.54020)]. They also can be used to construct a number of spaces \(Y\) for which the Ramsey type relation \(\omega^* \to (Y)^ 1_ 2\) holds, and others for which it fails; see [\textit{W. W. Comfort}, \textit{Akio Kato} and \textit{Saharon Shelah}, Ann. New York Acad. Sci., 1993].
scientific article; zbMATH DE number 515188
scientific article