Submission date: 12 November 2013.

Abstract:

Let G be an abelian group, and F a downward directed family of subsets of G. The finest topology T on G under which F converges to 0 has been described by I. Protasov and E. Zelenyuk. In particular, their description yields a criterion for T to be Hausdorff. They then show that if F is the filter of cofinite subsets of a countable subset X\subseteq G, there is a simpler criterion: T is Hausdorff if and only if for every g in G-{0} and positive integer n, there is an S in F such that g does not lie in the n-fold sum n(S\cup{0}\cup-S). In this note, their proof is adapted to a larger class of families F. In particular, if X is any infinite subset of G, κ any regular infinite cardinal ≤ Card(X), and F the set of complements in X of subsets of cardinality <κ, then the above criterion holds. We then give some negative examples, including a countable downward directed set F of subsets of Z not of the above sort which satisfies the “g notin n(S\cup{0}\cup-S)” condition, but does not induce a Hausdorff topology.

Comments: 10 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy.

Mathematics Subject Classification: 22A05 (Primary), 03E04, 54A20 (Secondary)

Keywords and phrases:

Full text arXiv 1311.2648: pdf, ps.