On group topologies determined by families of sets

Abstract

Let $G$ be an abelian group, and $F$ a downward directed family of subsets of $G$. The finest topology $\mathcal{T}$ 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 $\mathcal{T}$ 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: $\mathcal{T}$ 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 $G$, $κ$ any regular infinite cardinal $\leq\mathrm{card}(X)$, 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 $\mathbb{Z}$ not of the above sort which satisfies the "$g\notin n(S\cup\{0\}\cup-S)$" condition, but does not induce a Hausdorff topology. We end with a version of our main result for noncommutative $G$.