Sequences and dense sets

Abstract

By the classical Hewitt–Marczewski–Pondiczery theorem (see [2], [3]) the Tychonoff product of 2ω many separable spaces is separable [2,3].We consider the problem of the existence in the Tychonoff product of 2ω many separable spaces a dense countable subset Q, which contains no nontrivial convergent sequences.We prove (Theorem 4.1) that such dense set exists in the product ∏α∈2ωZα of separable spaces {Zα:α∈2ω} if in every space Zα (α∈2ω) there are two closed disjoint not empty sets, we call such spaces decomposable.The class of decomposable spaces includes not single point T1-spaces, but also some T0-spaces and some spaces, which are not even T0-spaces.