Fuzzy Points and Membership

One of the least satisfactory areas in the early development of fuzzy topology has been that surrounding the concept of fuzzy point. In the original classical theory, where values are taken in the closed unit interval I it soon became apparent that, in order to build up a reasonable theory, points should be defined as fuzzy singletons while membership requires strict inequality. So crisp points, taking value 1, are excluded, and fuzzy topology would seem not to include general topology. This disturbing state of affairs was to some extent overcome by [Pu and Liu 1980a] who replaced membership by quasi-coincidence (not belonging to the complement, where belonging is taken as ≤), thus reinstating crisp points. More recently [Hu 1985] has drawn attention to a duality between quasi-coincidence and strict inequality membership. The duality, however, is only partial [Warner 1989]. Details of the various definitions and corresponding neighbourhood theories are clearly described by [Kerre and Ottoy 1987].