© 1996, Topology Atlas

An ordered topological space is a topological space $(X, T)$ equipped with a partial order $<=$. Usual compatibility conditions between the topology and order include convexity ($T$ has a basis of order-convex sets) and the T_2-ordered property ($<=$ is closed in the product $X$ times $X$). Many of the rudiments of ordered spaces were compiled in Leopoldo Nachbin's classic Topology and Order, which appeared in Portuguese in 1950 and in English 15 years later. The 1979 survey by T. H. Choe [Partially Ordered Topological Spaces, An. Acad. brasil. Cienc.,(1979) 51,(1) 53-63] provides bibliographical references to that date.

An ordered space $(X^*, T^*, <=^*)$ is an ordered compactification of $(X, T, <=)$ if $(X^*, T^*)$ is a compactification of $(X, T)$ and $<=^*$ extends $<=$. Since every topological space $(X, T)$ can be considered as a trivially ordered space $(X, T, =)$, the theory of ordered spaces and ordered compactifications includes the usual topological theory as a special case. Other important special cases, each with its own set of techniques, include the totally ordered spaces and lattices. The types of questions considered for topological compactifications, such as the lattice structure of compactifications of a given space, construction and properties of compactifications, and consideration of remainders $X^*$ minus $X$, are still relevant. The general partially ordered cases are frequently more difficult than the corresponding topological cases. For example, there is currently no known filter construction for the ordered version of the Stone-Cech compactification.

Ordered spaces and ordered compactifications have been studied extensively in terms of quasi-uniformities and proximities, and are increasingly used in theoretical computer science.