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Abstract

With a slight modification of a previous argument due to Schechter, we show that the Axiom of Choice is equivalent to the following topological statement: “If a product of a non-empty family of sets is closed in a topological (Tychonoff) product, then at least one of the factors is closed”. We also discuss the case on which one adds the hypothesis that the closed product of sets is a non-empty set.

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We show that the cardinality of power homogeneous T 5 compacta X is bounded by 2 c ( X ) . This answers a question of J. van Mill, who proved this bound for homogeneous T 5 compacta. We further extend some results of I. Juhász, P. Nyikos and Z. Szentmiklóssy and as a corollary we prove that consistently every power homogeneous T 5 compactum is first countable. This improves a theorem of R. de la Vega who proved this consistency result for homogeneous T 5 compacta.

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Abstract  

This is a sequel of the work done on (strongly) monotonically monolithic spaces and their generalizations. We introduce the notion of monotonically κ-monolithic space for any infinite cardinal κ and present the relevant results. We show, among other things, that any σ-product of monotonically κ-monolithic spaces is monotonically κ-monolithic for any infinite cardinal κ; besides, it is consistent that any strongly monotonically ω-monolithic space with caliber ω 1 is second countable. We also study (strong) monotone κ-monolithicity in linearly ordered spaces and subspaces of ordinals.

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