Displaying similar documents to “Partition properties of ω1 compatible with CH”

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...

Almost all submaximal groups are paracompact and σ-discrete

O. Alas, I. Protasov, M. Tkačenko, V. Tkachuk, R. Wilson, I. Yaschenko (1998)

Fundamenta Mathematicae

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We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.

Chains and antichains in Boolean algebras

M. Losada, Stevo Todorčević (2000)

Fundamenta Mathematicae

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We give an affirmative answer to problem DJ from Fremlin’s list [8] which asks whether M A ω 1 implies that every uncountable Boolean algebra has an uncountable set of pairwise incomparable elements.

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).

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 .