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2 Bulletin of the Polish Academy of Sciences. Mathematics

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Cohen algebras and nowhere dense ultrafilters

100%

Błaszczyk A. , Szymański A.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

tom Vol. 49, no 1

15--25

We study a class of countable extremally disconnected spaces vrithout isolated points. We show that the Cech-Stone compactification of a space from this class admits a semi-open continuous maps onto a Cantor cube if and only if the space itself is modelled by an ultrafilter on omega that is not nowhere dense.

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

58%

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

tom Vol. 53, no 4

349-359

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 2R, 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).