Apartness spaces and uniform neighbourhood structures

Two notions of ‘neighbourhood structure’ are compared within a constructive framework, before a third, new notion is introduced: that of a pre-uniform neighbourhood structure. It is shown that with every basic uniform neighbourhood structure on an inhabited set there is associated a natural set–set apartness relation. A large class of uniform neighbourhood spaces is produced by a construction that classically gives the unique totally bounded uniform structure inducing the given apartness on a symmetric T1 apartness space with the Efremovič property. Thus although, constructively, it remains unknown (and very unlikely) that such an apartness is generally induced by a uniform structure, it closely corresponds with a uniform neighbourhood structure. Uniform neighbourhood structures also provide a setting for notions of total boundedness and continuity akin to those in uniform spaces.