On n-thin dense sets in powers of topological spaces

Abstract

A subset of a product of topological spaces is called n-thin if every its two distinct points differ in at least n coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable T_3 space X without isolated points such that X^n contains an n-thin dense subset, but X^{n+1} does not contain any n-thin dense subset.

We also observe that part of the construction can be carried out under MA.

Keywords

• dense set
• thin set
• κ-thin set
• independent family