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Universal spaces for classes of scattered eberlein compact spaces
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- Journal:
- The Journal of Symbolic Logic / Volume 71 / Issue 3 / September 2006
- Published online by Cambridge University Press:
- 12 March 2014, pp. 1073-1080
- Print publication:
- September 2006
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- Article
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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 δκof 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 δκ such that every member of δκ 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
. Assuming that 
= ω1, we prove that there exists a space X ∈ which is universal in the sense of embeddings. We also show that it is consistent that there exists a space X bΕ, universal in the sense of embeddings, but δω1 does not contain an universal element in the sense of mappings onto.
. Assuming that 
= 