Rajkov remainder and other group-remainders of a topological group

Abstract

If G is a dense subgroup of a topological group B, then the subspace X=B∖G is called a group-remainder, or a g-remainder, for short, of the topological group G, and B is said to be a group-extension of G. In this paper, g-remainders of topological groups are studied. We show that if X is a Lindelöf g-remainder of a topological group G, and X contains a nonempty compact subset of countable character in X, then G, X, and the Rajkov completion of G are Lindelöf p-spaces; in addition, any g-remainder of G is a Lindelöf p-space (Theorem 4.2). It follows that a topological group G is a Lindelöf p-space provided some nonempty g-remainder of it is a Lindelöf p-space. If some nonempty g-remainder of G is a paracompact p-space, then the topological group G is a paracompact p-space as well (Theorems 3.5 and 3.6).