A NOTE ON LOCALLY COMPACT REGIONALLY COMPACT-BY-ABELIAN GROUPS

Abstract

If X is a class of locally compact groups, a locally compact group G is called regionally X if each compact subset of G is contained in a closed X-subgroup of G. The class of regionally X groups is denoted by RX. In this paper, some general results about the structure of regionally [FD] groups are established, where [FD] denotes the class of locally compact groups G such that the topological commutator subgroup D(G) is compact. For example, if the quotient group GZ(G) modulo the center of G is regionally compact then G is regionally [FD]. On the other hand, we also show that G R[FD] is equivalent to the fact that any nite subset of G is contained in a [FD] subgroup. As an application we prove that the set compn (G) consisting of the n-tuples of elements of G contained in a common compact subgroup is a closed subgroup of the cartesian product Gn, for each positive integer n.