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Universal spaces for classes of scattered eberlein compact spaces

Published online by Cambridge University Press:  12 March 2014

Murray Bell  and
Witold Marciszewski
Murray Bell
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, R3T 2N2, Canada
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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 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.

Keywords

compact spacescatteredEberleinUniform Eberleinuniversal
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 3 , September 2006 , pp. 1073 - 1080
Copyright
Copyright © Association for Symbolic Logic 2006

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