Topological Group Foundations of Rigid Space Geometry

Abstract

1. Hilbert, after building up geometry from a point of view which relegated considerations to the background [4], built up plane geometry afresh [5] on the foundation of groups of homeomorphisms of the number plane, both continuity concepts. It is this latter point of view with which we are concerned in this paper. Hilbert carried out this program only for the plane but he hinted that it might be possible to carry it out in a somewhat similar way for three-space. Kerekjarto took up the problem [6] for threespace and made a great deal of progress with it, but, as he wrote before the recent developments in topological groups, he found it necessary to employ a stronger set of axioms than is necessary now. Relying on the theory of topological groups we recently characterized the rotation group of three-space [9], and in commenting on that work P. A. Smith [12] suggested that it might provide the basis for an extension of Hilbert's program to three-space. The purpose of this paper is twofold. In the first place we shall characterize the classical space geometries on the basis of a fairly weak set of axioms, and in the second place we shall show that Hilbert's axioms for the case of the plane can be weakened by replacing what might be called his three-point condition by a two-point condition. We achieve this latter purpose more or less incidentally to the first. In comparing our set of three-space axioms with Hilbert's axfoms for the plane we find that the first axiom is the same in both cases. The third axiom of this paper is weaker than Hilbert's, and our second purpose above is to show that this weaker axiom also suffices in Hilbert's case. Our second axiom is weaker than Hilbert's second axiom in that it relates to the subgroup leaving a single point fixed, but it is incomparably stronger in what it asks of that one subgroup. We do not, in this paper, settle the question of whether or not Hilbert's second axiom is adequate for three-space geometries. This question is bound up with an unsolved problem concerning transformation groups. Finally we wish to remark that instead of assuming that the space we are dealing with is ordinary three-space, it is only necessary to make certain topological assumptions on the space from which it follows by virtue of the