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 $G$, $\kappa$ any regular infinite
cardinal $\leq\mathrm{card}(X)$, and $F$ the set of complements in $X$ of
subsets of cardinality $<\kappa$, 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
"$g
otin n(S\cup\{0\}\cup-S)$" condition, but does not induce a Hausdorff
topology.
We end with a version of our main result for noncommutative $G$.
