Two-ended topological groups

Abstract

Introduction. Let G be a locally compact, connected topological group (satisfying the second countability axiom). Let G* be a compact space which contains a dense subset G' homeomorphic to the space G and is such that G*-G' is totally disconnected. Then, Freudenthal has proved [1, Satz 1X, p. 277]1 that the set G*-G' consists of at most two distinct points. Actually, Freudenthal's theorem even for topological groups is more general than here stated, and this theorem is an application to group spaces of a wider theory of ends of topological spaces. However, we shall quote only so much of Freudenthal's results as are necessary to this paper. It will be convenient to regard G' as identical with G so that G is topologically imbedded in G*. We shall call a locally compact, connected group G two-ended if a G* exists such that G* G consists of two distinct points. The simplest example of such a group is the additive group of reals. Other examples are afforded by the direct product of this group and any compact connected topological group; it is likely that these are the only examples. The principal objective of this note is the following theorem.