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When are order scattered and topologically scattered the same? (English) Zbl 0553.06007

Orders: description and roles, Proc. Conf. Ordered sets appl., l’Arbresle/France 1982, Ann. Discrete Math. 23, 61-80 (1984).
[For the entire collection see Zbl 0539.00003.]
A subset A of a poset P is said to be order dense if for a,b\(\in A\) with \(a<b\) there is some \(c\in A\) such that \(a<c<b\). P is order scattered if every nonempty order dense subset of P is an antichain. A topological space is topologically scattered if every nonempty subspace has an isolated point with respect to the relative topology. The author discusses the connection between the concepts of order scattered and topologically scattered in various settings in which both a topological and an order theoretic structure are present. The author develops the ideas of this paper, more completely in a paper to appear in Houston J. Math.
Reviewer: A.R.Stralka

MSC:

06B05 Structure theory of lattices
06B30 Topological lattices
54F99 Special properties of topological spaces
06A06 Partial orders, general

Citations:

Zbl 0539.00003