A Characterization of Order Topologies by Means of Minimal T0-Topologies

In this article we give a purely topological characterization for a topology J on a set X to be the order topology with respect to some linear order R on X, as follows. A topology J on a set X is an order topology$\operatorname{iff} (X, \mathfrak J)$is a T1-space and J is the least upper bound of two minimal T0-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, J) is homeomorphic to the reals with the usual topology$\operatorname{iff} (X, \mathfrak J)$is a connected, separable, T1-space, and J is the least upper bound of two noncompact minimal T0-topologies [Theorem 2].