Ordered Compactifications, Galois Connections, and Quasi-uniformities

I would like to thank the organizers of the UNISA Topology Workshop, Koena Rufus Nailana and Sergio Salbany, for doing an excellent job acquiring funding, making arrangements, and especially for assembling an enthusiastic group of participants. An ordered topological space (X, τ,≤) is a set X with a topology τ and a partial order ≤. We usually assume some forms of compatibility between the topology and order, such as convexity of the topology (τ has a base of ≤-convex sets) or the T2-ordered condition (the graph of ≤ is closed in the product (X, τ) × (X, τ)). We assume both these conditions for all ordered spaces considered here. The study of ordered topological spaces may be considered more general than the study of topological spaces, for any topological space (X, τ) may be thought of as an ordered space (X, τ,=) trivially ordered by equality. The classical introduction to ordered topological spaces is Leopoldo Nachbin’s Topology and Order [12], written in Portuguese in 1950 and translated into English in 1965. Besides the classical approach to ordered spaces which I usually use, Brümmer, Künzi, Nailana, Salbany, and others have used more sophisticated approaches, including bitopological techniques [13, 16], quasi-uniform techniquess [14, 9], nonstandard analysis techniques [17], and categorical techniques [2]. An ordered compactification of an ordered topological space (X, τ,≤) is a compact T2-ordered space (X ′, τ ′,≤′) such that (X, τ) is dense in (X ′, τ ′) and the restriction of ≤′ to X agrees with ≤. The classical study of compactifications of topological spaces amounts to the study of trivially ordered compactifications of trivially ordered topological spaces. Richard Chandler’s Hausdorff Compactifications [3] provides an excellent introduction to theory of topological compactification. Only the completely regular topological