On pseudocompactness of remainders of topological groups and some classes of mappings

Abstract

We investigate, when a topological group G is pseudocompact at infinity, that is, when bG⧹G is pseudocompact, for each compactification bG of G . It is established that if G is a non-discrete topological group such that every compact subspace of G is scattered, then G is metrizable if and only if it is not pseudocompact at infinity. More generally, a topological group in which every compact subspace has a point of countable character is either metrizable or pseudocompact at infinity. In particular, each non-metrizable topological group of cardinality at most ω 1 is pseudocompact at infinity, while it need not be countably compact at infinity even in the countable case. We observe that every pseudocompact group is pseudocompact at infinity. None of these results generalizes to the class of all Tychonoff spaces. On the other hand, every P -space is shown to be countably compact at infinity. We also introduce a curious concept of a crowded mapping, unifying those of a condensation and of an open continuous mapping with a dense-in-itself image. From the general results obtained it follows that if a countable space Y is an image of a dense-in-itself metrizable space under a condensation, then Y is not countably compact at infinity. The importance of the notion of a crowded mapping for the theory of topological groups may lie in the following fact established in Section 3: every continuous homomorphism of one topological group into another with a non-open kernel is crowded.