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; $\kappa $-thin set; independent family
DOI: DOI 10.14712/1213-7243.2015.148
AMS Subject Classification: 54B10 54A35