TOPOLOGY WITHOUT POINTS

Abstract

Other theories that develop topology without points are either excessively artificial or suffer from a lack of rigor. In this paper it is assumed that worlds W are composed of parts that form a complete Boolean algebra [xbar] and that the collection [Wbar] of all points of W is a certain subcollection of all filters defined over [xbar]. Two axioms are given for points which, given suitable definitions, convert [Wbar] into a compact Hausdorff space. Nearness collections of parts of W are defined which satisfy all the axioms of Herrlich for nearness except that closure is defined without mentioning points and consequently one may define closed and open parts. A category of worlds is defined in which the objects are lattices of closed parts of a world and the arrows are roughly speaking the far-preserving mps. It is shown that the category of compact T₁-spaces is a reflective subcategory of the category of worlds.