On dense subsets, convergent sequences and projections of Tychonoff products

Abstract

It is well know that the Tychonoff product of 2ω many separable spaces is separable [2,3].We consider for the Tychonoff product of 2ω many separable spaces the problem of the existence of a dense countable subset, which contains no nontrivial convergent in the product sequences.The first result was proved by W.H. Priestley. He proved [14] that such dense set exists in the Tychonoff product ∏α∈2ωIα of closed unit intervals.We prove (Theorem 3.2) that such dense set exists in the Tychonoff product ∏α∈2ωZα of 2ω many Hausdorff separable not single point spaces.We prove that in ∏α∈2ωZα there is a countable dense set Q⊆∏α∈2ωZα such that for every countable subset S⊆Q a set πA(S) is dense in a face ∏α∈AZα for some A, |A|=ω.We prove (Theorem 3.4) that in ∏α∈2ωIα there is a countable set, that is dense but sequentially closed in ∏α∈2ωIα with the Tychonoff topology and is closed and discrete in ∏α∈2ωIα with the box topology (Theorem 3.4).