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Quarter-stratifiability in ordered spaces. (English summary) Proc. Amer. Math. Soc. 134 (2006), no. 6, 1835–1847 (electronic). A topological space (X, τ) is called quarter-stratifiable if there is a function g: {1, 2,...} × X → τ such that (1) for each n the collection {g(n, x): x ∈ X} covers X, and (2) if y ∈ g(n, xn) for each n, then the sequence 〈xn 〉 converges to y [T. O. Banakh, Mat. Stud. 18 (2002), no. 1, 10–28; MR1968755 (2004d:54023)]. If in this definition the condition (1) is replaced by the requirement that x ∈ g(n, x) for each n and x, a characterization of semi-stratifiable spaces [G. D. Creede, Pacific J. Math. 32 (1970), 47–54; MR0254799 (40 #8006)] results. A generalized ordered space, or GO-space, is a linear ordered set endowed with a topology τ that is finer than the order topology [D. J. Lutzer, Dissertationes Math. Rozprawy Mat. 89 (1971), 32 pp.; MR0324668 (48 #3018)]. The aim of the paper under review is to investigate quarter-stratifiable GO-spaces. Various properties of these spaces are collected. A characterization of the quarter-stratifiable spaces follows. It is shown that the class of quarter-stratifiable spaces is hereditary. The theory is illustrated by several examples. Reviewed by J. M. Aarts Reference

Year: 2013

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