On dense subsets, convergent sequences and projections of Tychonoff products

It is well know that the Tychonoff product of 2(omega) many separable spaces is separable [2,3]. We consider for the Tychonoff product of 2(omega) 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 Pi(alpha is an element of 2 omega) I-alpha of closed unit intervals. We prove (Theorem 3.2) that such dense set exists in the Tychonoff product Pi(alpha is an element of 2 omega) Z(alpha) of 2(omega) many Hausdorff separable not single point spaces. We prove that in Pi(alpha is an element of 2 omega) Z(alpha) there is a countable dense set Q subset of Pi(alpha is an element of 2 omega) Z(alpha) such that for every countable subset S subset of Q a set pi(A) (S) is dense in a face Pi(alpha is an element of 2 omega) Z(alpha) for some A, vertical bar A vertical bar = omega. We prove (Theorem 3.4) that in Pi(alpha is an element of 2 omega) I-alpha there is a countable set, that is dense but sequentially closed in Pi(alpha is an element of 2 omega) I-alpha with the Tychonoff topology and is closed and discrete in Pi(alpha is an element of 2 omega) I-alpha with the box topology (Theorem 3.4). (C) 2017 Elsevier B.V. All rights reserved.