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

Abstract

In this article we give a purely topological characterization for a topology ℑ \Im on a set X X to be the order topology with respect to some linear order R R on X X , as follows. A topology ℑ \Im on a set X X is an order topology iff ( X , ℑ ) (X,\Im ) is a T 1 {T_1} -space and ℑ \Im is the least upper bound of two minimal T 0 {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 , ℑ ) (X,\Im ) is homeomorphic to the reals with the usual topology iff ( X , ℑ ) (X,\Im ) is a connected, separable, T 1 {T_1} -space, and ℑ \Im is the least upper bound of two noncompact minimal T 0 {T_0} -topologies [Theorem 2].