A universal continuum of weight $\aleph $

Dow, Alan; Hart, Klaas Pieter

doi:10.1090/s0002-9947-00-02601-5

Cited by 22 publications

(24 citation statements)

References 18 publications

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“…They give three proofs of this fact, each of which relies on model-theoretic notions in some essential way. The proof of our main theorem is most similar to their third proof, found in Section 3 of [9].…”

Section: Outline Of the Proofsupporting

confidence: 56%

“…In Section 5, we will show that both Parovičenko's theorem about continuous images of * and the Dow-Hart theorem about continuous images of H * can be derived as relatively straightforward corollaries of our main theorem. In light of this, it is unsurprising that our proof uses some of the same ideas found in [5] and [9].…”

Section: Outline Of the Proofmentioning

confidence: 97%

“…Our use of elementarity is inspired by the work of Dow and Hart in [9], where they prove that every continuum of weight ℵ 1 is a continuous image of H * , the Stone-Čech remainder of H = [0, ∞). They give three proofs of this fact, each of which relies on model-theoretic notions in some essential way.…”

Section: Outline Of the Proofmentioning

confidence: 99%

“…We are now in a position to prove the main theorem. As mentioned in the introduction, our proof technique parallels that in Section 3 of Dow and Hart's article [9]. In order to make things easier for the reader (especially the reader already familiar with [9]), we have tried to match our notation to that of [9] wherever possible.…”

Section: Continuous If and Only If For Every Open Cover U Of X There mentioning

confidence: 99%

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ABSTRACT ω-LIMIT SETS

Brian¹

2018

J. symb. log.

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The shift map σ on ω* is the continuous self-map of ω* induced by the function n ↦ n + 1 on ω. Given a compact Hausdorff space X and a continuous function f : X → X, we say that (X, f) is a quotient of (ω*, σ) whenever there is a continuous surjection Q : ω*→ X such that Q ○ σ = σ ○ f.Our main theorem states that if the weight of X is at most ℵ1, then (X, f) is a quotient of (ω*, σ), if and only if f is weakly incompressible (which means that no nontrivial open U ⊆ X has $f\left( {\bar{U}} \right) \subseteq U$). Under CH, this gives a complete characterization of the quotients of (ω*, σ) and implies, for example, that (ω*, σ−1) is a quotient of (ω*, σ).In the language of topological dynamics, our theorem states that a dynamical system of weight ℵ1 is an abstract ω-limit set if and only if it is weakly incompressible.We complement these results by proving (1) our main theorem remains true when ℵ1 is replaced by any κ < p, (2) consistently, the theorem becomes false if we replace ℵ1 by ℵ2, and (3) OCA + MA implies that (ω*, σ−1) is not a quotient of (ω*, σ).

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Section: Outline Of the Proofsupporting

confidence: 56%

Section: Outline Of the Proofmentioning

confidence: 97%

Section: Outline Of the Proofmentioning

confidence: 99%

Section: Continuous If and Only If For Every Open Cover U Of X There mentioning

confidence: 99%

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ABSTRACT ω-LIMIT SETS

Brian¹

2018

J. symb. log.

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“…The latter also hinges on the fortuitous fact that the theory of atomless Boolean algebras admits elimination of quantifiers. See also [8], where similar model-theoretic methods were applied to the lattice of closed subsets of a Stone-Čech remainder.…”

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confidence: 99%

Rigidity of Continuous Quotients

Farah

Shelah

2014

J. Inst. Math. Jussieu

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We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand-Naimark duality we conclude that the assertion that the Stone-Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.The present paper has two largely independent parts moving in two opposite directions. The first part ( § § 1-4) uses model theory of metric structures, and it is concerned with the degree of saturation of various reduced products. The second part ( § 5) uses set-theoretic methods and it is mostly concerned with rigidity of Stone-Čech 1 remainders of locally compact Polish spaces. (A topological space is Polish if it is separable and completely

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On universal Banach spaces of density continuum

Brech

Koszmider

2011

Isr. J. Math.

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Abstract. We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density not bigger than continuum isomorphically embeds into X (called a universal Banach space of density c). It is well known that ℓ∞/c 0 is such a space if we assume the continuum hypothesis. However, some additional set-theoretic assumption is needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density c. Thus, the problem of the existence of a universal Banach space of density c is undecidable using the usual axioms of set-theory.We also prove that it is consistent that there are universal Banach spaces of density c, but ℓ∞/c 0 is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0, c]) into ℓ∞/c 0 .

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A universal continuum of weight $\aleph $

Cited by 22 publications

References 18 publications

ABSTRACT ω-LIMIT SETS

ABSTRACT ω-LIMIT SETS

Rigidity of Continuous Quotients

On universal Banach spaces of density continuum

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