Sequential order of product spaces (Q1902983)

This paper investigates the sequential order of the products of sequential spaces. The following are main results in this paper. Here, for a sequential space \(X\), let \(so (X)\) be the sequential order of \(X\). Also, for a space \(X\), let \(S(X)\) be the sequential coreflection of \(X\); that is, \(S[X]\) is a new space having a sequential closure topology: \(F \subset X\) is closed in \(X\) iff \(C \cap F\) is closed in \(C\) for any compact metric subset \(C\) of \(X\). All spaces are assumed to be Hausdorff. (A) Let \(X\) be a sequential space, and let \(Y\) be a regular (locally) countably compact sequential space. Then \(X \times Y\) is sequential, and \(so (X \times Y) \leq so (X) + so (Y)\). (B) Let \(X\) be a sequential space, and let \(Y\) be a first countable space. Then \(so (S [X \times Y]) \leq so (X) + 1\). (C) Let \(X\) and \(Y\) be Fréchet spaces with point-countable \(k\)- networks. If \(X \times Y\) is sequential, then \(so (X \times Y) \leq 2\). (D) (CH) There exist Fréchet spaces \(X\) and \(Y\) such that \(X \times Y\) is a sequential space with \(so (X \times Y) = \omega_1\).