On group topologies determined by families of sets

G. M. Bergman
University of California, Berkeley, USA
Let $G$ be an abelian group, and $F$ a downward directed family of subsets of $G.$ In \cite{P+Z}, I. Protasov and E. Zelenyuk describe the finest group topology $\mathcal{T}$ on $G$ under which $F$ converges to $0;$ in particular, their description yields a criterion for $\mathcal{T}$ to be Hausdorff. They then show that if $F$ is the filter of cofinite subsets of a countable subset $X\subseteq G$ (the Fr\'{e}chet filter on $X),$ there is a simpler criterion: $\mathcal{T}$ 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 $G,$ $\kappa$ any regular infinite cardinal $\leq\mathrm{card}(X),$ and $F$ the set of complements in $X$ of subsets of cardinality less than $\kappa,$ then the above criterion holds. We also give some negative examples, including a countable downward directed set $F$ (not of the above sort) of subsets of $\mathbb{Z}$ which satisfies the ''$g\notin n(S\cup\{0\}\cup -S)$'' condition but does not induce a Hausdorff topology.
group topology; family of sets; Hausdorff topology
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