On dense subsets, convergent sequences and projections of Tychonoff products (Q1680130)

The author extends his work in [Topology Appl. 221, 300--308 (2017; Zbl 1379.54009)] to prove that in a product of continuum many non-trivial separable Hausdorff spaces one can find a countable dense set~\(Q\) such that whenever \(S\) is an infinite subset of~\(Q\) the projection of~\(S\) onto some countable subproduct is dense. In addition, the Tychonoff cube \([0,1]^\mathbb{R}\) is shown to have a countable dense subset as above that is closed and discrete in the box topology.