Apartness, Topology, and Uniformity: a Constructive View
Douglas Bridges, Peter M. Schuster, Luminiţa Vı̂ţă · Mathematical logic quarterly · 2002
The theory of apartness spaces, and their relation to topological spaces (in the point–set case) and uniform spaces (in the set–set case), is sketched. New notions of local decomposability and regularity are investigated, and the latter is used to produce an example of a classically metrisable apartness on R that cannot be induced constructively by a uniform structure.