A concept of nearness

Abstract This paper offers solutions for two problems which have attracted many topologists over the years: 1. (1) It provides a natural and reasonably simple concept of “nearness” which unifies various concepts of “topological structures” in the sense that the category Near of all nearness spaces and nearness preserving maps contains the categories (a) of all topological Ro-spaces and continuous maps, (b) of all uniform spaces and uniformly continuous maps (Weil [34], Turkey [33]), (c) of all proximity spaces and δ-maps (Efremovic [9], Smirnov [28,29]), (d) of all contiguity spaces and contiguity maps (Ivanova and Ivanov [17]) as nicely embedded (either bireflective or bicoreflective) full (!) subcategories. 2. (2) It provides a general method by means of which as many T1-extension of a T1-space can be obtained as might be reasonably expected; namely, all strict extensions (in the sense of Banaschewski [3]).