Universal spaces for classes of scattered Eberlein compact spaces

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 𝒮_κ 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 𝒮_κ such that every member of 𝒮_κ 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 𝒮_ω₁. Assuming that 𝔡= ω₁, we prove that there exists a space X∈𝒮_ω₁, which is universal in the sense of embeddings. We also show that it is consistent that there exists a space X∈𝒮_ω₁, universal in the sense of embeddings, but 𝒮_ω₁ does not contain an universal element in the sense of mappings onto.