Displaying similar documents to “The sequential topology on complete Boolean algebras”

Bohr compactifications of discrete structures

Joan Hart, Kenneth Kunen (1999)

Fundamenta Mathematicae

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We prove the following theorem: Given a⊆ω and 1 ≤ α < ω 1 C K , if for some η < ℵ 1 and all u ∈ WO of length η, a is Σ α 0 ( u ) , then a is Σ α 0 .We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: Σ 1 1 -Turing-determinacy implies the existence of 0 .

Nonseparable Radon measures and small compact spaces

Grzegorz Plebanek (1997)

Fundamenta Mathematicae

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We investigate the problem if every compact space K carrying a Radon measure of Maharam type κ can be continuously mapped onto the Tikhonov cube [ 0 , 1 ] κ (κ being an uncountable cardinal). We show that for κ ≥ cf(κ) ≥ κ this holds if and only if κ is a precaliber of measure algebras. Assuming that there is a family of ω 1 null sets in 2 ω 1 such that every perfect set meets one of them, we construct a compact space showing that the answer to the above problem is “no” for κ = ω. We also give alternative...

The measure algebra does not always embed

Alan Dow, Klaas Hart (2000)

Fundamenta Mathematicae

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The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(ℕ)/fin. We also discuss errors in previous results on the embeddability of the measure algebra.

Embedding partially ordered sets into ω ω

Ilijas Farah (1996)

Fundamenta Mathematicae

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We investigate some natural questions about the class of posets which can be embedded into ⟨ω,≤*⟩. Our main tool is a simple ccc forcing notion H E which generically embeds a given poset E into ⟨ω,≤*⟩ and does this in a “minimal” way (see Theorems 9.1, 10.1, 6.1 and 9.2).

The concept of boundedness and the Bohr compactification of a MAP Abelian group

Jorge Galindo, Salvador Hernández (1999)

Fundamenta Mathematicae

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Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an ℒ ∞ -group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification....

Properties of the class of measure separable compact spaces

Mirna Džamonja, Kenneth Kunen (1995)

Fundamenta Mathematicae

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We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular)...

Expansions of the real line by open sets: o-minimality and open cores

Chris Miller, Patrick Speissegger (1999)

Fundamenta Mathematicae

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The open core of a structure ℜ := (ℝ,<,...) is defined to be the reduct (in the sense of definability) of ℜ generated by all of its definable open sets. If the open core of ℜ is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of ℝ is finite or uncountable, or if ℜ defines addition and multiplication and every definable open subset of ℝ has finitely many connected components, then the open core...

On absolutely divergent series

Sakaé Fuchino, Heike Mildenberger, Saharon Shelah, Peter Vojtáš (1999)

Fundamenta Mathematicae

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We show that in the ℵ 2 -stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš’ result that under c f ( ) = the two algebras are isomorphic [15].

On absolute retracts of ω*

A. Bella, A. Błaszczyk, A. Szymański (1994)

Fundamenta Mathematicae

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An extremally disconnected space is called an absolute retract in the class of all extremally disconnected spaces if it is a retract of any extremally disconnected compact space in which it can be embedded. The Gleason spaces over dyadic spaces have this property. The main result of this paper says that if a space X of π-weight ω 1 is an absolute retract in the class of all extremally disconnected compact spaces and X is homogeneous with respect to π-weight (i.e. all non-empty open sets...