A characterization of order topologies by means of minimal ₀-topologies

In this article we give a purely topological characterization for a topology 5 on a set X to be the order topology with respect to some linear order R on X, as follows. A topology 5 on a set X is an order topology iff (X, 3) is a T1-space and 5 is the least upper bound of two minimal To-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, 3) is homeomorphic to the reals with the usual topology iff (X, 3) is a connected, separable, T1-space, and 5 is the least upper bound of two noncompact minimal To-topologies [Theorem 2 ].