Chapter 28 - Reflection of Topological Properties to ℵ1

Abstract

This chapter discusses the reflection of topological properties to א1. Large cardinals exhibit reflection phenomena—if some universal property holds below κ, it holds everywhere; alternatively, if there is an object with some property, there is one of size less than κ. A standard set-theoretic technique is to collapse a large cardinal κ to be small, for example, א1, א2, or 2א0, and to see if some particular instance of reflection holds. The prototypical example of this is that if a super compact cardinal is Lévy-collapsed to w2, then for every regular cardinal λ ≥w2 and every stationary set S of ω-cofinal ordinals in λ , there is an α <λ of cofinality ω1 such that S ∩ α is stationary in α. A simplest, interesting question—due to Fleissner—is the following problem: “Is every first countable א1‑collection-wise Hausdorff space collection-wise Hausdorff?” There are easy counterexamples if “first countable” is omitted. More interesting, but apparently harder to deal with is Hamburger's.