Suitable sets for subgroups of direct sums of discrete groups

Let G = circle plus (x<tau) G(x) be the direct surn of (discrete) groups endowed with the linear group topology whose base at the identity consists of the subgroups U-x = circle plus(xless than or equal tobeta<tau) G(beta), alpha<tau. We show that every subgroup H of G has a generating suitable set. In addition, if H is not closed in G, then H has a closed generating suitable set. The case of topologically orderable topological groups is considered. It is proved that if a topologically orderable group G has a (closed) Suitable set, then every dense subgroup of G also has a (closed) suitable set. (C) 2002 Elsevier Science B.V. All rights reserved.