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COMPLETE CCC BOOLEAN ALGEBRAS, THE ORDER SEQUENTIAL TOPOLOGY, AND A PROBLEM Of Von Neumann
, 2005
"... Let B be a complete ccc Boolean algebra and let τs be the topology on B induced by the algebraic convergence of sequences in B. 1. Either there exists a Maharam submeasure on B or every nonempty open set in (B,τs) is topologically dense. 2. It is consistent that every weakly distributive complete cc ..."
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Let B be a complete ccc Boolean algebra and let τs be the topology on B induced by the algebraic convergence of sequences in B. 1. Either there exists a Maharam submeasure on B or every nonempty open set in (B,τs) is topologically dense. 2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure. 3. The topological space (B,τs) is sequentially compact if and only if the generic extension by B does not add independent reals. Examples are also given of ccc forcings adding a real but not independent reals.
MAHARAM ALGEBRAS AND COHEN REALS
, 2007
"... We show that the product of any two nonatomic Maharam algebras adds a Cohen real. As a corollary of this and a result of Shelah (1994) we obtain that the product of any two nonatomic ccc Souslin forcing notions adds a Cohen real. ..."
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We show that the product of any two nonatomic Maharam algebras adds a Cohen real. As a corollary of this and a result of Shelah (1994) we obtain that the product of any two nonatomic ccc Souslin forcing notions adds a Cohen real.
G la̧b, Ideals with bases of unbounded Borel complexity
 MLQ Math. Log. Q
"... Abstract We present several naturally defined σideals which have Borel bases but, unlike for the classical examples, these bases are not of bounded Borel complexity. We investigate settheoretic properties of such σideals. ..."
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Abstract We present several naturally defined σideals which have Borel bases but, unlike for the classical examples, these bases are not of bounded Borel complexity. We investigate settheoretic properties of such σideals.
Maharam Algebras
, 2008
"... Maharam algebras are complete Boolean algebras carrying a positive continuous submeasure. They were introduced and studied by Maharam in [24] in relation to Von Neumann’s problem on the characterization of measure algebras. The question whether every Maharam algebra is a measure algebra has been the ..."
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Maharam algebras are complete Boolean algebras carrying a positive continuous submeasure. They were introduced and studied by Maharam in [24] in relation to Von Neumann’s problem on the characterization of measure algebras. The question whether every Maharam algebra is a measure algebra has been the main open problem in this area for around 60 years. It was finally resolved by Talagrand [31] who provided the first example of a Maharam algebra which is not a measure algebra. In this paper we survey some recent work on Maharam algebras in relation to the two conditions proposed by Von Neumann: weak distributivity and the countable chain condition. It turns out that by strengthening either one of these conditions one obtains a ZFC characterization of Maharam algebras. We also present some results on Maharam algebras as forcing notions showing that they share some of the well known properties of measure algebras.
ON THE UNIQUENESS OF MEASURE AND CATEGORY σIDEALS ON 2ω
, 2006
"... Abstract. We prove that if a pair 〈I, J 〉 of ccc, translation invariant σideals on 2ω has the Fubini Property, then I = J. This leads to a slightly improved exposition of a part of the FarahZapletal proof of an invariant version of their theorem which characterizes the measure and category σideal ..."
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Abstract. We prove that if a pair 〈I, J 〉 of ccc, translation invariant σideals on 2ω has the Fubini Property, then I = J. This leads to a slightly improved exposition of a part of the FarahZapletal proof of an invariant version of their theorem which characterizes the measure and category σideals on 2ω as essentially the only ccc definable σideals with Fubini Property. 1.
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"... Electronic supplementary material for “On the social nature of eyes: The effect of social cues in interaction and individual choice tasks.” ..."
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Electronic supplementary material for “On the social nature of eyes: The effect of social cues in interaction and individual choice tasks.”