# Order topology

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The topology on a linearly ordered set , with linear order , which has a base consisting of all possible intervals of .

Here "interval" is used in the sense of "open interval" , i.e. a set of the form where (or possibly and/or ). The order topology may be considered on partially ordered as well as linearly ordered sets; on a linearly ordered set it coincides with the interval topology which has the closed intervals as a subbase for the closed sets, but in general it is different. On a complete linearly ordered set, the order topology is characterized by order convergence: that is, a net (see Generalized sequence) converges to a point if and only if there are an increasing net and a decreasing net , indexed by the same directed set , such that for all and .

How to Cite This Entry:
Order topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Order_topology&oldid=15011
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article