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

100%
Keremedis K.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
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tom 53
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nr 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 $2^{ℝ}$, 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 ).
2
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On BPI Restricted to Boolean Algebras of Size Continuum

64%
Hall E., Keremedis K.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
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tom 61
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nr 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 $2^{𝓟(ω)}$ the restriction 𝓑|Y = {Y ∩ B: B ∈ 𝓑} of the standard clopen base 𝓑 of $2^{𝓟(ω)}$ 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.
3
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Remarks on the Stone Spaces of the Integers and the Reals without AC

51%
Herrlich H., Keremedis K., Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2011
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tom 59
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nr 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 $2^{𝓟(X)}$, where 2 is 2 = {0,1} 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.
4
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On the Compactness and Countable Compactness of $2^{ℝ}$ in ZF

51%
Keremedis K., Felouzis E., Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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tom 55
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nr 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 "$2^{ℝ}$ is countably compact" and "$2^{ℝ}$ is compact"
5
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Countable Compact Scattered T₂ Spaces and Weak Forms of AC

51%
Keremedis K., Felouzis E., Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
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tom 54
|
nr 1
75-84
EN
We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable. (3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space is scattered. (4) It is not provable in ZF+¬AC that there exists a countable compact T₂ space which is dense-in-itself.
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