Sequential properties of lexicographic products

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 X n has first and last elements, the lexicographic product ∏ n = 1 ∞ X n is sequentially connected if and only if each X n is. Whenever each X n is a LOTS without first and last elements, then it is shown that ∏ n = 1 ∞ X n is always a sequential space. The lexicographic product ∏ α < ω 1 X α , where ω 1 is the first uncountable ordinal, is also investigated and it is shown that if each X α contains at least two points, then ∏ α < ω 1 X α 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 X α , α < ω 1 , does not have first and last elements, we show that the lexicographic product ∏ α < ω 1 X α is a P-space without isolated points.