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• W2073945915 abstract "In a complete lattice it is possible to define a notion of convergence (for arbitrary nets) known as order convergence (o-convergence) ; for definitions see [l,p.5°]and [3, p. 65]. As a general rule o-convergence is not a topological convergence; i.e., the lattice cannot be topologized so that nets o-converge if and only if they converge with respect to the topology [2]. It is of interest to know when these two types of convergence coincide. A number of questions could be posed here, but we shall deal only with the question of when a topological space can be suitably embedded in a complete lattice. To make this statement more precise we shall give the following definition. Definition. A topological space is said to be an O-space if it is homeomorphic to a subset s0 of a complete lattice s and if every net in fio converges (with respect to the topology for s0) to a limit in s0 if and only if it o-converges to this limit. For example, every completely regular Hausdorff space is an O-space because it is homeomorphic to a subset of the direct product of unit intervals and if this direct product is partially ordered componentwise, then it becomes a complete lattice in which o-convergence is the same as convergence with respect to the product topology. In this paper we prove the following theorem." @default.
• W2073945915 created "2016-06-24" @default.
• W2073945915 creator A5037108360 @default.
• W2073945915 date "1965-04-01" @default.
• W2073945915 modified "2023-09-24" @default.
• W2073945915 title "Order convergence and topological convergence" @default.
• W2073945915 cites W2039554731 @default.
• W2073945915 doi "https://doi.org/10.1090/s0002-9939-1965-0178449-3" @default.
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