Sequential properties of lexicographic products

Abstract

In this article, using the characterization of almost P-points of a linearly ordered topological space (LOTS) in terms of sequences, we observe that in the category of linearly ordered topological spaces, quasi F-spaces and almost P-spaces coincide. This coincidence gives examples of quasi F-spaces with no F-points. We also use the characterization of sequentially connected LOTS in terms of almost P-points to show that whenever each LOTS Xn has first and last elements, the lexicographic product ∏n=1∞Xn is sequentially connected if and only if each Xn is. Whenever each Xn is a LOTS without first and last elements, then it is shown that ∏n=1∞Xn is always a sequential space. The lexicographic product ∏α<ω1Xα, where ω1 is the first uncountable ordinal, is also investigated and it is shown that if each Xα contains at least two points, then ∏α<ω1Xα is always an almost P-space (a quasi F-space) but it is neither sequential nor sequentially connected. Using this lexicographic product, we give an example of a quasi F-space in which the set of F-points and the set of non-F-points are dense. Whenever each Xα, α<ω1, does not have first and last elements, we show that the lexicographic product ∏α<ω1Xα is a P-space without isolated points.