2000
DOI: 10.1090/s0002-9939-00-05403-4
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Universal uniform Eberlein compact spaces

Abstract: Abstract. A universal space is one that continuously maps onto all others of its own kind and weight. We investigate when a universal Uniform Eberlein compact space exists for weight κ. If κ = 2 <κ , then they exist whereas otherwise, in many cases including κ = ω 1 , it is consistent that they do not exist. This answers (for many κ and consistently for all κ) a question of Benyamini, Rudin and Wage of 1977.

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Cited by 12 publications
(18 citation statements)
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“…In [3] M. Bell showed that assuming κ ω = κ there is a universal uniform Eberlein compact space of weight κ. M. Fabian, G. Godefroy and V. Zizler showed that the class of Banach spaces satisfying the hypothesis of Proposition 1.2 corresponding to the class of uniform Eberlein compact spaces is the class of UG Banach spaces. So, these two results imply by Proposition 1.2 that there are universal UG Banach spaces of density c = ω 1 .…”
Section: Introductionmentioning
confidence: 99%
“…In [3] M. Bell showed that assuming κ ω = κ there is a universal uniform Eberlein compact space of weight κ. M. Fabian, G. Godefroy and V. Zizler showed that the class of Banach spaces satisfying the hypothesis of Proposition 1.2 corresponding to the class of uniform Eberlein compact spaces is the class of UG Banach spaces. So, these two results imply by Proposition 1.2 that there are universal UG Banach spaces of density c = ω 1 .…”
Section: Introductionmentioning
confidence: 99%
“…Therefore X continuously maps onto every nonempty closed subspace of an x an. It follows from Theorem 4.1 of Bell [5] PROOF. Let P be the forcing that adds a>i Cohen reals to V. If there exists X e §", n V F which can be mapped continuously onto every Y € § m D V p , then by Proposition 3.1 there exists a universal graph of cardinality co\ in V p ; but this is not so-Kojman and Shelah [13].…”
Section: Let a B And C Be Objects If I: A -» B Is An Embedding Anmentioning
confidence: 94%
“…We refer to Fleischer's [10,11] non-relational reformulation of the generalization by Morley and Vaught [15] of Jonsson [12]. For the interested reader, versions of this construction can be found in Bell and Slomson [4], or Chang and Keisler [7] (see also the proof of Theorem 3.3 in [5]).…”
Section: Let a B And C Be Objects If I: A -» B Is An Embedding Anmentioning
confidence: 99%
See 1 more Smart Citation
“…Another context in which universal objects have arisen is the study of Polish groups and, more generally, Polish spaces, see e.g., (Uspenskii, 1986;Ben-Yaacov, 2014). There has been considerable interest in the question of existence of universal objects among Banach spaces, and more generally, metric and topological spaces of a particular form, e.g., (Bourgain, 1980;Katětov, 1988;Bell, 2000;Džamonja, 2006;Brech and Koszmider, 2012;Brech and Koszmider, 2013;Džamonja, 2014). Finally, universal objects appear also in the context of sofic groups, see e.g., (Pestov, 2008;Thomas, 2010).…”
Section: The Test Question: Universality and The Universality Ordermentioning
confidence: 99%