A characterization of order topologies by means of minimal 𝑇₀-topologies

Thron, W. J.; Zimmerman, Susan J.

doi:10.1090/S0002-9939-1971-0283767-7

A characterization of order topologies by means of minimal $T_{0}$-topologies
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by W. J. Thron and Susan J. Zimmerman PDF

Proc. Amer. Math. Soc. 27 (1971), 161-167 Request permission

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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