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Preprint Number 655
655. George M. Bergman On group topologies determined by families of sets E-mail: 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 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: |

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