Displaying similar documents to “ -representation and set-prolongations”

The number of L κ -equivalent nonisomorphic models for κ weakly compact

Saharon Shelah, Pauli Vaisanen (2002)

Fundamenta Mathematicae

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For a cardinal κ and a model M of cardinality κ let No(M) denote the number of nonisomorphic models of cardinality κ which are L , κ -equivalent to M. We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ¹₁-definable over V κ . By [SV] it is possible to have a generic extension where the possible numbers of equivalence...

Class preserving mappings of equivalence systems

Ivan Chajda (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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By an equivalence system is meant a couple 𝒜 = ( A , θ ) where A is a non-void set and θ is an equivalence on A . A mapping h of an equivalence system 𝒜 into is called a class preserving mapping if h ( [ a ] θ ) = [ h ( a ) ] θ ' for each a A . We will characterize class preserving mappings by means of permutability of θ with the equivalence Φ h induced by h .

Indiscernibles and dimensional compactness

C. Ward Henson, Pavol Zlatoš (1996)

Commentationes Mathematicae Universitatis Carolinae

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This is a contribution to the theory of topological vector spaces within the framework of the alternative set theory. Using indiscernibles we will show that every infinite set u S G in a biequivalence vector space W , M , G , such that x - y M for distinct x , y u , contains an infinite independent subset. Consequently, a class X G is dimensionally compact iff the π -equivalence M is compact on X . This solves a problem from the paper [NPZ 1992] by J. Náter, P. Pulmann and the second author.

Interpolation of κ -compactness and PCF

István Juhász, Zoltán Szentmiklóssy (2009)

Commentationes Mathematicae Universitatis Carolinae

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We call a topological space κ -compact if every subset of size κ has a complete accumulation point in it. Let Φ ( μ , κ , λ ) denote the following statement: μ < κ < λ = cf ( λ ) and there is { S ξ : ξ < λ } [ κ ] μ such that | { ξ : | S ξ A | = μ } | < λ whenever A [ κ ] < κ . We show that if Φ ( μ , κ , λ ) holds and the space X is both μ -compact and λ -compact then X is κ -compact as well. Moreover, from PCF theory we deduce Φ ( cf ( κ ) , κ , κ + ) for every singular cardinal κ . As a corollary we get that a linearly Lindelöf and ω -compact space is uncountably compact, that is κ -compact for all uncountable cardinals...

On equivalence relations second order definable over H(κ)

Saharon Shelah, Pauli Vaisanen (2002)

Fundamenta Mathematicae

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Let κ be an uncountable regular cardinal. Call an equivalence relation on functions from κ into 2 second order definable over H(κ) if there exists a second order sentence ϕ and a parameter P ⊆ H(κ) such that functions f and g from κ into 2 are equivalent iff the structure ⟨ H(κ), ∈, P, f, g ⟩ satisfies ϕ. The possible numbers of equivalence classes of second order definable equivalence relations include all the nonzero cardinals at most κ⁺. Additionally, the possibilities are closed...

Open maps do not preserve Whyburn property

Franco Obersnel (2003)

Commentationes Mathematicae Universitatis Carolinae

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We show that a (weakly) Whyburn space X may be mapped continuously via an open map f onto a non (weakly) Whyburn space Y . This fact may happen even between topological groups X and Y , f a homomorphism, X Whyburn and Y not even weakly Whyburn.