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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A characterization of order topologies by means of minimal $ T\sb{0}$-topologies


Authors: W. J. Thron and Susan J. Zimmerman
Journal: Proc. Amer. Math. Soc. 27 (1971), 161-167
MSC: Primary 54.56; Secondary 06.00
MathSciNet review: 0283767
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Abstract: In this article we give a purely topological characterization for a topology $ \Im $ on a set $ X$ to be the order topology with respect to some linear order $ R$ on $ X$, as follows. A topology $ \Im $ on a set $ X$ is an order topology iff $ (X,\Im )$ is a $ {T_1}$-space and $ \Im $ is the least upper bound of two minimal $ {T_0}$-topologies [Theorem 1 ]. From this we deduce a purely topological description of the usual topology on the set of all real numbers. That is, a topological space $ (X,\Im )$ is homeomorphic to the reals with the usual topology iff $ (X,\Im )$ is a connected, separable, $ {T_1}$-space, and $ \Im $ is the least upper bound of two noncompact minimal $ {T_0}$-topologies [Theorem 2].


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0283767-7
PII: S 0002-9939(1971)0283767-7
Keywords: Lattice of topologies, pre-order relation, complete order, dense order, connected space, separable space, nested topology, minimal $ {T_0}$-topology
Article copyright: © Copyright 1971 American Mathematical Society