On \(n\)-thin dense sets in powers of topological spaces. (Q2798070)

A subset of a product of topological spaces is called \(n\)-thin if any two distinct points of it differ in at least \(n\) coordinates. For any natural number \(n\geq2\) the author provides, under the continuum hypothesis, a countable Hausdorff regular space \(X\) without isolated points such that \(X^n\) admits a dense \(n\)-thin subset, but \(X^{n+1}\) does not admit any dense \(n\)-thin subset. Under Martin's axiom a weaker conclusion is obtained, namely that \(X^{n+1}\) does not admit any dense \((n+1)\)-thin subset. The main tool is a method of constructing topologies from carefully chosen independent families of sets of natural numbers.