A Topological Approach to Geometry

Abstract

The purpose of this paper is to summarize a method by which at least certain topological spaces can be characterized up to homeomorphism class. The method employs the notion of a geometry defined on a set by means of distinguished subsets called k-flats which are generalizations of k-dimensional subspaces. The set theoretic axioms defining a geometry are intended to embody the essence of anything that mathematicians have ever called a geometry. If the set on which a geometry has been defined is also a topological space, then the topology and geometry may be axiomatically related in such a manner that the topology can be completely characterized. We shall concern ourselves in this paper primarily with characterizing Rm as a topological space using the properties generalized from the Euclidean geometry on Rm. In Rm, the usual Euclidean geometry, the standard metric, and the algebraic structure of Rn as a vector space are tightly bound together. In our initial abstraction we do away with coordinatization, metrics, and the algebraic structure of geometries derived from linear manifolds. A geometry is defined as follows: Let X be a set. An element of X is called a point. We define G = { F-1, FO, ... , Fn } to be a geometry on X if the following axioms are satisfied: 1) Fi consists of subsets of X, I <i ?n. An element of Fi is called an i-flat, or merely a flat. 2) The only 1-flat is the empty set, i.e. F-1'{ 0}. 3) Fi consists of proper subsets of X, -1 i <n. 4) Every set of i+1 points not all contained in some k-flat, k<i, is contained in a unique i-flat, -1 i <s?n. 5) The intersection of any two flats is again a flat. 6) If k <i, then no i-flat is contained in any k-flat. n is called the length of G and we write 1(G) = n. i is called the dimension of Fi, as well as the dimension of any flat f in Ft; we write dim f=i. By G* we denote { J1, * * *, Fn, F'n+l }, where F'n+l = { X }. X is then considered to be an n+1-flat. The axioms defining a geometry are independent. The most trivial example of a geometry on a set X is where the i-flats of X are subsets of X of cardinality i+1. Clearly the usual Euclidean geometry on Rm is a geometry in our sense of length m -1. The great circle geometry on the 2-sphere S2 is also a geometry on the set of points of S2. If we let F-1 = { 0 }, FO { { x, y } I x is antipodal to y }, and Fl = { C| C is a great circle }, then G = { PF, FO, Fl } is a geometry of length 1. The length of any geometry is assumed to be finite throughout this paper, although this is not required for many results. Suppose now, and for the remainder of this paper, that X is a topological space on which a geometry of length m -1 _ 0 has been defined. We need a concept which will act as a link between geometry and topology, and for this purpose we use a generalized notion of convexity.