Thin-Type Dense Sets and Related Properties

We build on Gruenhage, Natkaniec, and Piotrowski’s study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that X has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (GC) but not witnessed by a collection of finite sets. We show that the question of when the topological sum of two countable strongly irresolvable spaces satisfies (NC) is related to the Rudin-Keisler order on βω. We also introduce and study the concepts of < κ-thin and superslim dense sets. 1 Background and Definitions The concepts of “thin” and “very thin” sets were defined by Piotrowski [6]. “Slim” was defined by Gruenhage in [4]. Definition 1.1. Let X = ∏ α<κXα be a product space, and let D ⊆ X. 1. D is thin if whenever x, y ∈ D with x 6= y, then |{α < κ : xα 6= yα}| > 1. 2. D is very thin if whenever x, y ∈ D with x 6= y, xα 6= yα for all α < κ. 3. D is slim if for every non-empty proper subset K ⊂ κ and ν ∈ ∏ α∈K Xα, the set D∩C(ν) is nowhere dense in C(ν), where C(ν) = {x ∈ X : x K = ν} is the cross-section of X at ν. We will call D ∩ C(ν) the cross-section of D at ν. AMS Subject classification (2000): Primary 54B10; Secondary: 54F05;03E75.