On resolvability of products

Juhász, István and Soukup, Lajos and Szentmiklóssy, Zoltán (2023) On resolvability of products. FUNDAMENTA MATHEMATICAE, 260. pp. 281-295. ISSN 0016-2736

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Abstract

All spaces below are T0 and crowded (i.e. have no isolated points). For n≤ω let M(n) be the statement that there are n measurable cardinals, and Π(n) (Π+(n)) that there are n+1 (0-dimensional T2) spaces whose product is irresolvable. We prove that M(1), Π(1) and Π+(1) are equiconsistent. For 1<n<ω we show that CON(M(n)) implies CON(Π+(n)). Finally, CON(M(ω)) implies the consistency of having infinitely many crowded 0-dimensional T2 spaces such that the product of any finitely many of them is irresolvable. These settle old problems of Malykhin. Concerning an even older question of Ceder and Pearson (1967), we show that the following are consistent modulo a measurable cardinal: (i) There is a 0-dimensional T2 space X with ω2≤Δ(X)≤2ω1 whose product with any countable space is not ω2-resolvable, hence not maximally resolvable. (ii) There is a monotonically normal space X with Δ(X)=ℵω whose product with any countable space is not ω1-resolvable, hence not maximally resolvable. These significantly improve a result of Eckertson (1997).

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