A topological group G is sequentially complete if it is sequentially closed in any other topological group. We show that for a Tychonoff space X, the free topological group F(X) is sequentially complete iff the free Abelian topological group A(X) is sequentially complete iff X is sequentially closed in βX. Furthermore, the free precompact Abelian group F(X,PA) is sequentially complete iff the space X is sequentially closed in βX. We consider also other forms of weak completeness, namely ω-completeness and k-completeness, introduced analogously by means of the ω-closure and the k-closure. We prove that the groups A(X) and F(X) are ω-complete (k-complete) iff X is ω-closed (k-closed) in the Dieudonné completion μX of X.

A topological group G is called sequentially complete if it is sequentially closed in any ther topological group (or equivalently, G is sequentially closed in its Raikov completion (G) over tilde). We establish the following compactness criterion in the class of connected Abelian groups of non-measurable size: a group in this class is compact iff it is minimal and sequentially complete. We also describe the structure of sequentially complete minimal Abelian groups in the general case. Coincidence of hereditary disconnectedness and zero dimensionality is established for various classes of sequentially complete groups. (C) 2001 Elsevier Science B.V. All rights reserved.

We discuss various generalisations of countable compactness for topological groups that are related to completeness. The sequentially complete groups form a class closed with respect to taking direct products and closed subgroups. Surprisingly, the stronger version of sequential completeness called sequential h-completeness (all continuous homomorphic images are sequentially complete) implies pseudocompactness in the presence of good algebraic properties such as nilpotency. We also study quotients of sequentially complete groups and find several classes of sequentially q-complete groups (all quotients are sequentially complete). Finally, we show that the pseudocompact sequentially complete groups are far from being sequentially q-complete in the following sense: every pseudocompact Abelian group is a quotient of a pseudocompact Abelian sequentially complete group.

A discrete subset S of a topological group G with identity 1 is called suitable for G if S generates a dense subgroup of G and S∪{1} is closed in G. We study various algebraic and topological conditions on a group G which imply the existence of a suitable set for G as well as the restraints imposed by the existence of such a set. The classes Sc, Sg and Scg of topological groups having a closed, generating and a closed generating suitable set are considered. The problem of stability of these classes under the product, direct sum operations and taking subgroups or quotients is investigated. We show that (totally) minimal Abelian groups often have a suitable set. It is also proved that every Abelian group endowed with the finest totally bounded group topology has a closed generating suitable set. More generally, the Bohr topology of every locally compact Abelian group admits a suitable set.

We investigate C-compact and relatively pseudocompact subsets of Tychonoff spaces with a special emphasis given to subsets of topological groups. It is shown that a relatively pseudocompact subset of a space X is C-compact in X, but not vice versa. If, however, X is a topological group, then these properties coincide. A product of two C-compact (relatively pseudocompact) subsets A of X and B of Y need not be C-compact (relatively pseudocompact) in X×Y, but if one of the factors X,Y is a topological group, then both C-compactness and relative pseudocompactness are preserved. We prove under the same assumption that, with A and B being bounded subsets of X and Y, the closure of A×B in υ(X×Y) is naturally homeomorphic to clυXA×clυYB, where υ stands for the Hewitt realcompactification. One of our main technical tools is the notion of an R-factorizable group. We show that an R-factorizable subgroup H of an arbitrary group G is z-embedded in G. This fact is applied to prove that the group operations of an R-factorizable group G can always be extended to the realcompactification υG of G, thus giving to υG the topological group structure. We also prove that a C-compact subset A of a topological group G is relatively pseudocompact in the subspace B=A·A−1·A of G.

If a discrete subset S of a topological group G with identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We construct in ZFC a Lindelöf topological group L such that t(L)·ψ(L)≤ℵ0 and L does not have a suitable set. We also give a ZFC example of a countably compact topological group H with no suitable set; in addition, the closure of every countable subset of H is compact. It is proved that a non-pseudocompact topological group with a dense strictly σ-discrete subset has a closed suitable set. This implies, in particular, that a free (Abelian) topological group on a metrizable space has a closed suitable set.

We present a concise survey of old and new results concerning cardinal functions on topological groups and then establish various relations between the classes of σ-compact, ℵ0-bounded and R-factorizable topological groups. The article is addressed to the general topology minded reader with no (or little) experience in topological algebra.