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dense set; thin set; $\kappa $-thin set; independent family
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.
[En] Engelking R.: General Topology. revised and completed edition, Sigma series in pure mathematics, 6, Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[GNP] Gruenhage G., Natkaniec T., Piotrowski Z.: On thin, very thin, and slim dense sets. Topology Appl. 154 (2007), no. 4, 817–833. DOI 10.1016/j.topol.2006.08.007 | MR 2294630
[HG] Hutchison J., Gruenhage G.: Thin-type dense sets and related properties. Topology Appl. 158 (2011), no. 16, 2174–2183. DOI 10.1016/j.topol.2011.07.005 | MR 2831904
[Je] Jech T.: Set Theory. The Third Millennium Edition, revised and expanded, Springer, Berlin, 2002. MR 1940513 | Zbl 1007.03002
[Pi] Piotrowski Z.: Dense subsets of product spaces. Questions Answers Gen. Topology 11 (1993), 313–320. MR 1234206
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