Tychonoff Products of Two-Element Sets and Some Weakenings of the Boolean Prime Ideal Theorem

Kyriakos Keremedis¹

¹ Department of Mathematics University of the Aegean 83 200 Karlovassi (Samos), Greece

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) no. 4, pp. 349-359.

Résumé

Let $X$ be an infinite set, and $\mathcal{P}(X)$ the Boolean algebra of subsets of $X$. We consider the following statements: BPI($X$): Every proper filter of $\mathcal{P}(X)$ can be extended to an ultrafilter. UF($X$): $\mathcal{P}(X)$ has a free ultrafilter. We will show in ZF (i.e., Zermelo–Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI($\omega$). (ii) The Tychonoff product $2^{\mathbb{R}}$, where $2$ is the discrete space $\{0,1\}$, is compact. (iii) The Tychonoff product $[0,1]^{\mathbb{R}}$ is compact. (iv) In a Boolean algebra of size $\leq|\mathbb{R}|$ every filter can be extended to an ultrafilter. We will also show that in ZF, UF($\mathbb{R}$) does not imply BPI($\mathbb{R}% $). Hence, BPI($\mathbb{R}$) is strictly stronger than UF($\mathbb{R}$). We do not know if UF($\omega$) implies BPI($\omega$) in ZF. Furthermore, we will prove that the axiom of choice for sets of subsets of $\mathbb{R}$ does not imply BPI($\mathbb{R}$) and, in addition, the axiom of choice for well orderable sets of non-empty sets does not imply BPI($\omega $).

Détail

DOI : 10.4064/ba53-4-1

Keywords: infinite set mathcal boolean algebra subsets consider following statements bpi every proper filter mathcal extended ultrafilter mathcal has ultrafilter zermelo fraenkel set theory without axiom choice following statements equivalent bpi omega tychonoff product mathbb where discrete space compact iii tychonoff product mathbb compact boolean algebra size leq mathbb every filter extended ultrafilter mathbb does imply bpi mathbb hence bpi mathbb strictly stronger mathbb know omega implies bpi omega furthermore prove axiom choice sets subsets mathbb does imply bpi mathbb addition axiom choice orderable sets non empty sets does imply bpi omega