Structure of locally compact groups with metrizable connected components up to negligible subsets

Abstract

Let G be a non-discrete, locally compact group and then Go the connected component containing the identity e. Let 2, 2 o be Haar measures on G, G o, respectively. This paper is mainly concerned with the topological and Haar measure-theoretic structure of certain G0-subsets in G and in coset spaces G/H. In particular, we will show that, after removal of a suitable meager F~-subset N, the topological structure of G becomes exceptionally simple if (and only if) Go is metrizable: G \ N is homeomorphic to the product of a discrete space, the space P of the irrationals, and a Cantor space (i. e., a product of copies of the discrete two-point space D). N can even be chosen so as to respect 20, the connected components of G, and the topological and algebraic structure of G; moreover, either N or G \ N can be made locally 2-negligible. As a consequence, in such a group G, almost all compact G0-subsets are homeomorphic with a Cantor space, and 2 is determined by Ga-subsets of Cantor type. Something similar can be done in a coset space G/H when one uses Skljarenko's topological description of G/H in [15]. I would like to thank the referee for his helpful comments. First, we fix some standing hypotheses, definitions, and notations: Throughout , H will be a closed non-open subgroup of G and then B 0 the connected component of the left coset space G/H containing H. As is well-known (cf. [15; p. 75]), B o = G O H/H. Let a (G) denote the smallest among all cardinal numbers of coset spaces G/F, with F an open subgroup of G for which F/Go is compact (cf. [t3; p. 54]); and let then S,(~) denote any discrete space of cardinality a (G). If X is a topological space, H (X) will stand for the collection of all zero-sets of X (i. e., sets ~0-1 (0) with continuous ~o: X ~ N). Then H c (X) will denote the set of all non-empty compact members of H (X); it will be equipped with