ON RIESZ THEOREM

Abstract

We prove a Riesz type criterion for a class of metric monoids: Local compactness implies finiteness of the Hausdorff dimension (and also of the topological dimension). We construct topological groups showing the necessity of some conditions. We finally prove that for some metric topological spaces finiteness of the algebraic dimension is equivalent to the finiteness of the Hausdorff dimension.

ACKNOWLEDGMENTS

I am grateful to the referee for his careful reading and pertinent remarks that contributed to correct some mistakes in the original preprint. The construction in theorem 4 was motivated by a question of the referee.