Formal Topology

The beginning of formal (or point-free topology) can be traced back to Whitehead (1915: revue de Metaphysique, ``Theorie relationnelle de l'espace''), and a survey of the state of the art in 1940 was written by Menger.

There is another older algebraic tradition: Dedekind and Weber (1882) analysed a purely algebraic presentation of Riemann surfaces where the notion of points is derived. This seems to have been inspired by the work of Kronecker. In this work, what are primary are polynomials with integer coefficients, and the solutions of these equations are considered to be ideal elements. Typically also, one does not work in one ``big'' algebraic closure given a priori, but one adds new symbolic elements that should satisfy some constraints (polynomial equations) when needed. We have started a formal presentation of the work of Dedekind and Weber in this work on valuations and this development. It is somewhat remarkable that the treatment of algebra in the XIXth century was in some sense logically more satisfactory than the one in the XXth century. The intensive use of Zorn's lemma in abstract algebra does not appear to be necessary.

The relevance of these concepts to constructive mathematics and computer science (especially domain theory) is that this gives a method to describe infinite objects (the points) in a purely finitary and algebraic way.

Last modified: Fri Sep 13 22:08:32 MET DST 2002