On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product $2^{\mathbb{R}}$

Eleftherios Tachtsis¹

¹ Department of Statistics and Actuarial-Financial Mathematics University of the Aegean Karlovassi, Samos, 83200, Greece

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 58 (2010) no. 2, pp. 91-107.

Résumé

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 (AC$^{\rm{WO}}$) does not imply “the Tychonoff product $2^\mathbb R$, 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 $2^\mathbb{R}$ is countably compact in ZF? 2. Assuming the Countable Axiom of Multiple Choice (CMC), the statements “every infinite subset of $2^{\mathbb{R}}$ has an accumulation point”, “every countably infinite subset of $2^{\mathbb{R}}$ has an accumulation point”, “$2^{\mathbb{R}}$ is countably compact”, and UF($\omega$) = “there is a free ultrafilter on $\omega$” are pairwise equivalent. 3. The statements “for every infinite set $X$, every countably infinite subset of $2^{X}$ has an accumulation point”, “every countably infinite subset of $2^{\mathbb{R}}$ has an accumulation point”, and UF($\omega$) are, in ZF, pairwise equivalent. Hence, in ZF, the statement “$2^{\mathbb{R}}$ is countably compact” implies UF($\omega$). 4. The statement “every infinite subset of $2^{\mathbb{R}}$ has an accumulation point” implies “every countable family of 2-element subsets of the powerset $\mathcal{P}(\mathbb{R})$ of $\mathbb{R}$ has a choice function”.5. The Countable Axiom of Choice restricted to non-empty finite sets, (CAC$_{\rm{fin}}$), is, in ZF, strictly weaker than the statement “for every infinite set $X$, $2^{X}$ is countably compact”.

Détail

DOI : 10.4064/ba58-2-1

Keywords: work set theory zermelo fraenkel set theory minus axiom choice following axiom choice well ordered families non empty sets does imply tychonoff product mathbb where discrete space countably compact answers negative following question keremedis felouzis tachtsis bull polish acad sci math does countable axiom choice families non empty sets reals imply mathbb countably compact assuming countable axiom multiple choice cmc statements every infinite subset mathbb has accumulation point every countably infinite subset mathbb has accumulation point mathbb countably compact omega there ultrafilter omega pairwise equivalent statements every infinite set every countably infinite subset has accumulation point every countably infinite subset mathbb has accumulation point omega pairwise equivalent hence statement mathbb countably compact implies omega statement every infinite subset mathbb has accumulation point implies every countable family element subsets powerset mathcal mathbb mathbb has choice function countable axiom choice restricted non empty finite sets cac fin strictly weaker statement every infinite set countably compact