Thin-type dense sets and related properties

We build on Gruenhage, Natkaniec, and Piotrowskiʼs 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 X2 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.