Sequences and dense sets (Q2293003)

In this paper the author calls a topological space \(X\) \textit{decomposable} if \(X\) contains two disjoint nonempty closed subsets (no separation axioms are assumed for \(X\)). Clearly, the class of decomposable spaces contains all \(T_1\)-spaces with at least two points and even many \(T_0\) and non-\(T_0\)-spaces. The main result of the paper is contained in the author's Theorem 4.1: Let \(\Pi_{\alpha\in 2^\omega} Z_\alpha\) be the Tychonoff product of separable decomposable spaces. Then there exists a countable dense set \(Q\subset \Pi_{\alpha\in 2^\omega} Z_\alpha\), which contains no convergent in \(\Pi_{\alpha\in 2^\omega} Z_\alpha\) nontrivial sequences. Theorem 4.1 generalizes some previous results of W. H. Priestley, P. Simon and the author with the same conclusion but stronger conditions on the spaces \(Z_\alpha\). Remark: The requirement that each \(Z_\alpha\) in the above theorem is a separable space is missing in the statement of Theorem 4.1 in the paper but is used in its proof and is mentioned in the abstract (and is clearly necessary for the conclusion of the theorem to be true).