Tychonoff products of two-element sets and some weakenings of the Boolean prime ideal theorem

• Autorzy: Keremedis K.

Identyfikatory

DOI

10.4064/ba53-4-1

• Języki publikacji: EN

Abstrakty

EN

Let X be an infinite set, and P(X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of P(X) can be extended to an ultrafilter. UF(X): 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(ω). (ii) The Tychonoff product 2^R, where 2 is the discrete space {0, 1}, is compact. (iii) The Tychonoff product [0, 1] ^R is compact. (iv) In a Boolean algebra of size ≤ |R| every filter can be extended to an ultrafilter. We will also show that in ZF, UF(R) does not imply BPI(R). Hence, BPI(R) is strictly stronger than UF(R). We do not know if UF(ω) implies BPI(ω) in ZF. Furthermore, we will prove that the axiom of choice for sets of subsets of R does not imply BPI(R) and, in addition, the axiom of choice for well orderable sets of non-empty sets does not imply BPI(ω).

Słowa kluczowe

EN

Boolean algebra; prime ideal; filter; ultrafilter; axiom of choice; weak axioms of choice; Tychonoff products

PL

algebra Boole'a; ideał pierwszy; filtr; filtr maksymalny; aksjomat wyboru

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2005
• Tom: Vol. 53, no 4
• Strony: 349--359
• Opis fizyczny: Bibliogr. 20 poz., tab.

Twórcy

autor

Keremedis K.

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

Bibliografia

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