Tolerance spaces: Origins, theoretical aspects and applications

This article considers the origins, theoretical aspects and applications of tolerance spaces. In terms of the origin of tolerance spaces, this article calls attention to the seminal work by J.H. Poincaré (1854–1912) and E.C. Zeeman (1925–) on establishing the foundations for tolerance spaces. During the period from 1895 to 1912, Poincaré introduced sets of sensations and sequences of almost the same sensations as a means of characterizing the physical spectrum. The perception of physical objects that are almost the same leads to a tolerance space view of visual perception as well as other forms of perception such as touch and sound. Roughly 60 years later (in 1962), Zeeman formally introduced the notion of a tolerance space as a useful means of investigating a geometry of visual perception. In addition to the general theory of tolerance spaces, this article also carries forward earlier work on perceptual tolerance relations and considers the resemblance (nearness) between tolerance spaces. From an information systems point of view, it can be observed that tolerance spaces have proved to be fruitful in a number of research areas. Evidence of the utility of tolerance spaces in information systems can be seen in the introduction of tolerance rough sets, tolerance approximation spaces, and tolerance near sets. The contribution of this article is an overview of tolerance spaces considered in the context of visual perception and a presentation of a formal basis for the study of perceptual tolerance spaces.