Science and Mathematics Faculty Publications

Thin-Type Dense Sets and Related Properties

Document Type

Article

Publication Date

10-1-2011

Journal Title

Topology and Its Applications

Volume

158

Issue

16

First Page

2174

Last Page

2183

Abstract

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 Formula Not Shown 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.

Keywords

Dense set; Thin set; Slim set; Superslim set; (GC); (NC); Ordered space

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