On sequentially h-complete groups

A topological group G is sequentially h-complete if all the continuous homomorphic images of G are sequentially complete. In this paper we give necessary and sufficient conditions on a complete group for being compact, using the language of sequential h-completeness. In the process of obtaining such conditions, we establish a structure theorem for ω-precompact sequentially h-complete groups. As a consequence we obtain a reduction theorem for the problem of c-compactness. All topological groups in this paper are assumed to be Hausdorff. A topological group G is sequentially h-complete if all the continuous homomorphic images of G are sequentially complete (i.e., every Cauchy-sequence converges). G is called precompact if for any neighborhood U of the identity element there exists a finite subset F of G such that G = UF . In [6, Theorem 3.6] Dikranjan and Tkačenko proved that nilpotent sequentially h-complete groups are precompact (also see [4]). Thus, if a group is nilpotent, sequentially h-complete and complete, then it is compact. Inspired by this result, the aim of this paper is to give necessary and sufficient conditions on a complete group for being compact, using the language of sequential h-completeness. This aim is carried out in Theorem 6. For an infinite cardinal τ , a topological group G is τ-precompact if for any neighborhood U of the identity element there exists F ⊂ G such that G = UF and |F | ≤ τ . In order to prove Theorem 6, we will first establish a strengthened version of the Guran’s Embedding Theorem for ω-precompact sequentially h-completely groups (Theorem 5). A topological groupG is c-compact if for any topological groupH the projection πH : G × H → H maps closed subgroups of G × H onto closed subgroups of H (see [12], [5] and [2], as well as [3]). The problem of whether every c-compact 2000 Mathematics Subject Classification. Primary 22A05, 22C05; Secondary 54D30. I gratefully acknowledge the financial support received from York University that enabled me to do this research. c ©0000 American Mathematical Society