On a topological construction of Juhasz and Shelah

Extract

In [2], Juhasz and Shelah use a forcing argument to show that it is consistent with GCH that there is a 0-dimensional T₂ topological space X of cardinality ℵ₃ such that every partition of the triples of X into countably many pieces has a nondiscrete (in the topology) homogeneous set. In this paper we will show how to construct such a space using a simplified (ω₂, 1)-morass with certain additional structure added to it. The additional structure will be a slight strengthening of a built-in ◊ sequence, analogous to the strengthening of ordinary ◊_k to ◊_S for a stationary set S ⊆ k.

Suppose 〈〈θ_α∣ ∝ ≤ ω₂〉, 〈_∝β∣α < β ≤ ω₂〉〉 is a neat simplified (ω₂, 1)-morass (see [3]). Let ℒ be a language with countably many symbols of all types, and suppose that for each α < ω₂, _α is an ℒ-structure with universe θ_α. The sequence 〈_α∣α < ω₂ is called a built-in ◊ sequence for the morass if for every ℒ-structure with universe ω₃ there is some α < ω₂ and some f ∈_αω2 such that f(_α) ≺ , where f(_α) is the ℒ-structure isomorphic to _α under the isomorphism f. We can strengthen this slightly by assuming that _α is only defined for α ∈ S, for some stationary set S ⊆ ω₂. We will then say that is a built-in ◊ sequence on levels in S if for every ℒ-structure with universe ω₃ there is some α ∈ S and some f ∈ _αω2 such that f(_α) ≺ .