Erik Palmgren and Peter Schuster

The theory of formal spaces and the more recent theory of apartness spaces have a priori not much more in common than that each of them was initiated as a constructive approach to general topology. We nonetheless try to do the first steps in relating these competing theories to each other. Formal topology was put forward in the mid 1980s by Sambin [11] in order to make available to Martin–Löf’s type theory [9] the concepts of classical topology that are worth keeping to such a constructive and predicative framework. The development of formal topology was inspired by, among other things, the theory of formal spaces worked out by Fourman and Grayson [8]. Since its early days formal topology has proved a fairly universal setting for doing topology in a point–free way. We refer to [12] for a recent and exhaustive survey of formal topology. The theory of apartness spaces was started by Bridges and Vı̂ţă [4] nearly twenty years later to reformulate set–theoretic topology as an extension of Bishop’s constructive analysis [2, 3]. The subsequent development of the theory of apartness spaces has also shed some light on its classical counterpart, the theory of proximity or nearness spaces. A comprehensive overview will be available soon [5]. In formal topology ‘basic neighbourhood’ is a primitive concept, whereas ‘point’ is a derived notion; as sets of basic neighbourhoods, points have to be handled with particular care to meet the needs of a predicative framework like Martin–Löf type theory. In the theory of apartness spaces, it is the other way round: as in classical topology, points are given as such, and (basic) neighbourhoods are sets of points. Since, however, it is hard to detect any truly impredicative move in the practice of Bishop’s constructive mathematics in general, we dare to undertake the following attempt to link formal topology and the theory of apartness spaces to each other. 1. Basic Definitions We recall the standard definitions associated with formal topologies and morphisms between them (approximable mappings). Definition 1.1. Let A be a set, and let be a relation between elements of A and subsets of A, i.e. ⊆ A× P(A). Extend to a relation between subsets of A by setting U V if and only if a V for all a ∈ U . For a preorder (A,≤) and a subset U ⊆ A, its downwards closure U≤ consists of those a ∈ A such that a ≤ b for some b ∈ U . 1991 Mathematics Subject Classification Primary 03F65, Secondary 03F60, 06D22, 54A05, 54E17, 54E05.