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On the existence of almost disjoint and MAD families without AC

Tachtsis Eleftherios

Bulletin of the Polish Academy of Sciences. Mathematics

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2019

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Vol. 67, no. 2

101--124

EN

In set theory without the Axiom of Choice (AC), we investigate the deductive strength and mutual relationships of the following statements: 1) Every infinite set X has an almost disjoint family A of infinite subsets of X with [formula]. (2) Every infinite set X has an almost disjoint family A of infinite subsets of X with [formula]. (3) For every infinite set X, every almost disjoint family in X can be extended to a maximal almost disjoint family in X. (4) For every infinite set X, no infinite maximal almost disjoint family in X has cardinality [formula]. (5) For every infinite set A, there is a continuum sized almost disjoint family A ⊆ Aω. (6) For every free ultrafilter U on ω and every infinite set A, the ultrapower Aω/U has cardinality at least [formula].

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On BPI Restricted to Boolean Algebras of Size Continuum

Hall E., Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2013

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Vol. 61, no 1

9--21

EN

(i) The statement P(ω)=“every partition of R has size ≤|R|” is equivalent to the proposition R(ω)=“for every subspace Y of the Tychonoff product 2P(ω) the restriction B|Y={Y∩B:B∈B} of the standard clopen base B of 2P(ω) to Y has size ≤|P(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of P(ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤|R| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of size ≤|R| has an ultrafilter.

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Remarks on the Stone Spaces of the Integers and the Reals without AC

Herrlich H., Keremedis K., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2011

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Vol. 59, no 2

101--114

EN

In ZF, i.e., the Zermelo–Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2P(X), where 2 is 2 = f0; 1g with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X =ω,R. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

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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

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2010

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Vol. 58, no 2

91-107

EN

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".

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On the compactness and countable compactness of 2R in ZF

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 4

293-302

EN

In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements "2R is countably compact" and "2R is compact".

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

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2005

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Vol. 53, no 4

349-359

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 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).