Homogenity versus finiteness

The relationship between basic physical assumptions as homogeneity of space on the one hand and local finiteness of the world on the other hand, is investigated. It is proved that there is sort of uncertainly relation between homogeneity and local finiteness: Geometric results, well known for Euclidian spaces as the Jordan theorem for simple curves are not true in finite homogeneous spaces. But for a weakened definition of homogeneity all important topological results remain true for locally finite topological spaces.