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On the set-theoretic strength of countable compactness of the Tychonoff product 2R

Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

Vol. 58, no 2

91-107

We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets (ACWO) does not imply "the Tychonoff product 2R, where 2 is the discrete space {0,1}, is countably compact" in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets of reals imply 2R is countably compact in ZF? 2. Assuming the Countable Axiom of Multiple Choice (CMC), the statements "every infinite subset of 2R has an accumulation point", "every countably infinite subset of 2R has an accumulation point", "2R is countably compact", and UF(ω) = "there is a free ultrafilter on ω" are pairwise equivalent. 3. The statements "for every infinite set X, every countably infinite subset of 2X has an accumulation point", "every countably infinite subset of 2R has an accumulation point", and UF(ω) are, in ZF, pairwise equivalent. Hence, in ZF, the statement "2R is countably compact "implies UF(ω). 4. The statement "every infinite subset of 2R has an accumulation point" implies "every countable family of 2-element subsets of the powerset Ρ(R) of R has a choice function". 5. The Countable Axiom of Choice restricted to non-empty finite sets, (CACfin), is, in ZF, strictly weaker than the statement "for every infinite set X, 2X is countably compact".