Einbettung von topologischen Raumgeometrien auf R3 in den reellen affinen Raum

Abstract

Betten [1] had defined topological spatial geometries on R 3: In R 3 a system L of closed subsets homeomorphic to R (the lines) and a system ℰ of closed subsets homeomorphic to R 2 (the planes) are given such that through any two different points passes exactly one line and through any three non-collinear points passes exactly one plane. Furthermore, ℒ and ℰ carry topologies such that the operations of joining and intersection are continuous. It is proved that any topological spatial geometry on R 3 can be imbedded into R 3 as an open convex subset K such that the lines in ℒ (planes in ℰ) are mapped onto intersections of lines (planes) of R 3 with K. The collineation group of the geometry is isomorphic to the subgroup of the colineation group of real projective space consisting of the automorphisms that map K into itself. In particular, it is a Lie group of dimension ⩽12.