Resolvable, not maximally resolvable spaces
According to \textit{J. G. Ceder} [Fundam. Math. 55, 87-93 (1964; Zbl 0139.40401)] following E. Hewitt (1963), a space \(X= (X,{\mathcal T})\) is said to be \(\kappa\)-resolvable if \(X\) is the union of \(\kappa\)-many pairwise disjoint dense subsets; and \(X\) is maximally resolvable if \(X\) is \(\Delta(X)\)-resolvable, where \(\Delta(X)= \min\{| U|: \emptyset\neq U\in{\mathcal T}\}\). \textit{E. K. van Douwen} [Topology Appl. 51, No. 2, 125-139 (1993; Zbl 0845.54028)] found (for fixed \(n<\omega\)) a Tikhonov space which is \(n\)-resolvable but not \((n+1)\)-resolvable, but the following question, though studied since 1967 by many workers, remains unsolved in ZFC: Is there an \(\omega\)-resolvable Tikhonov space which is not maximally resolvable? The present author describes ``a source of reasonable candidates for ZFC examples'', and he constructs explicit examples in models which admit a crowded, strong \(P_\kappa\), hereditarily irresolvable space; according to work of K. Kunen, A. Szymański and F. Tall, the existence of such models is equiconsistent with the existence of an uncountable measurable cardinal. With such a space in hand, the author in addition can answer negatively certain other longstanding questions in this area, as follows: (a) Must a product be maximally resolvable if one factor is crowded and maximally resolvable? (b) Must each open subset of a crowded, maximally resolvable space be maximally resolvable? (c) Must the union of crowded, maximally resolvable spaces be maximally resolvable? The construction technique introduced here by the author gives some new information even in ZFC, for example Corollary 4.2: For each \(\kappa\geq \omega\) there is a Tikhonov space that is a union of \(2^{2^\kappa}\)-many pairwise disjoint dense irresolvable subspaces, each of cardinality \(\kappa\).
scientific article; zbMATH DE number 1054141
scientific article