J. Group Theory 7 (2004), 255–264 Journal of Group Theory ( de Gruyter 2004 Hereditarily s-groups Desmond Robbie and Mikhail Tkachenko (Communicated by S. A. Morris) Abstract. We call a topological group G an s-group if G has a suitable set (the concept introduced by Hofmann and Morris), and an hs-group if every subgroup of G is an s-group. Similarly, cs-groups and gs-groups are topological groups having a closed suitable set and a generating suitable set, respectively. This gives rise (by the obvious mnemonic rule) to hcs- and hgs-groups, as well as hgcs-groups. We show that closed continuous homomorphisms preserve the classes of hs-, hcs-, hgs-, and hgcs-groups. This result is applied to prove that every compact hs-group is metrizable while every almost metrizable hs-group is metrizable. 1 Introduction A subset S of a topological group G with identity e is said to be a suitable set if it inherits the discrete topology from G, it is a closed subset of G nfeg, and the subgroup generated by S is dense in G. This notion was introduced by Hofmann and Morris in [11] and extensively studied in [6], [7], [8], [14], [16], [17]. It is known that all locally compact groups and all metrizable groups have a suitable set [11], [4], but neither Lindelo ¨f nor countably compact groups need have suitable sets [6]. Topological groups with closed and/or generating suitable sets were considered in [4], [7]. These articles contain a number of results concerning products and quotients of groups with (closed or generating) suitable sets. Our aim here is to consider topological groups all of whose subgroups have suit- able sets. The groups with this property are called hs-groups. If every subgroup of a topological group G has a closed (generating, or closed and generating) suitable set, then we call G an hcs-group (hgs-group or hcgs-group, respectively). It is shown in Lemma 2.3 that continuous closed homomorphisms preserve the classes of hs-, hcs-, hgs- and hcgs-groups. This result plays an important role in the proof of the fact that every compact hs-group is second countable (see Theorem 3.8). In its turn, the latter result implies that all almost metrizable hs-groups are metrizable (Corollary 3.9). Theorem 3.8 makes use of some facts from the C p -theory which are supplied with brief outlines of proofs. We also make essential use of several results from the struc- ture theory of compact topological groups. Unauthenticated | 74.112.201.150 Download Date | 2/11/13 5:46 PM