First and second category Abelian groups with then-adic topology
Edgar Howard · Pacific Journal of Mathematics · 1966
Throughout this paper the word group shall mean Abelian group.The w-adic topology of a group G is formed by taking the subgroups k !G as a base for the neighborhood system of the identity where & is a nonnegative integer.In this paper we list some properties of first and second category groups with the w-adic topology (a group is of first category if it is a countable union of nowhere dense sets).We characterize first and second category groups and prove the following: THEOREM: A torsion group is of second category if and only if G = H®D where H is bounded and D is divisible.THEOREM: Every torsion homomorphic image of a second category (e.g.complete) group is the direct sum of a bounded group and a divisible group.THEOREM: If G is reduced and of second category and G -Σ Ga 9 then there exists an integer n such that nG a -0 for all but finitely many a. THEOREM: If T is torsion, T is isomorphic to the torsion subgroup of a second category group.The notation and terminology will be essentially that of L. Fuchs in [1].The topological notations will be those of [6].We note the following.1.If A is a subset (subgroup) of B we write Aξ^B (A g B).2. ζxy denotes the cyclic group generated by x.3. + means sum not necessarily direct, and 0 means direct sum.4. By a first (second) category group G we shall mean that G is of first (second) category with the %-adic topology.5.The term " closed " will be used in the topological sense.2* On first and second category groups* In this section we shall use the facts that homomorphisms are continuous maps in the w-adic topology, and "onto" homomorphisms are open maps in this topology.The proof of the following is routine.LEMMA 2.1.If f is a homomorphism from G onto H, then