Cardinalities of locally compact groups and their stone-Cech compactifications

摘要

If G is any Hausdorff topological group and βG is its Stone-Cech compactification, then |G|≤|βG|≤2~(2|G|), where |G| denotes the cardinality of G. It is known that if G is a discrete group then |βG|=2~(2|G|) and if G is the additive group of real numbers with the Euclidean topology, then |βG|=2~(2|G|). In this paper the cardinality and weight of βG, for a locally compact group G, is calculated in terms of the character and Lindelof degree of G. The results make it possible to give a reasonably complete description of locally compact groups G for which |βG|=2~(2|G|) or even |βG|=|G|.