Abstract
We discuss the existence of universal spaces (either in the sense of embeddings or continuous images) for some classes of scattered Eberlein compacta. Given a cardinal κ, we consider the class 𝒮κ of all scattered Eberlein compact spaces K of weight ≤κ and such that the second Cantor-Bendixson derivative of K is a singleton. We prove that if κ is an uncountable cardinal such that κ = 2< κ, then there exists a space X in 𝒮κ such that every member of 𝒮κ is homeomorphic to a retract of X. We show that it is consistent that there does not exist a universal space (either by embeddings or by mappings onto) in 𝒮ω₁. Assuming that 𝔡= ω₁, we prove that there exists a space X∈𝒮ω₁, which is universal in the sense of embeddings. We also show that it is consistent that there exists a space X∈𝒮ω₁, universal in the sense of embeddings, but 𝒮ω₁ does not contain an universal element in the sense of mappings onto.
Citation
Murray Bell. Witold Marciszewski. "Universal spaces for classes of scattered Eberlein compact spaces." J. Symbolic Logic 71 (3) 1073 - 1080, September 2006. https://doi.org/10.2178/jsl/1154698593
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