On group topologies determined by families of sets 

Author 
gbergman@math.berkeley.edu
University of California, Berkeley, USA

Abstract 
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.

Keywords 
group topology; family of sets; Hausdorff topology

DOI 
doi:10.15330/ms.43.2.115128

Reference 
1. G. M. Bergman, On monoids, 2firs and semifirs, Semigroup Forum, 89 (2014), 293–335, http://arxiv.org/
abs/1309.0564.
2. P.M. Cohn, Universal algebra. – Second edition, Mathematics and its Applications, 6. D. Reidel Publishing Co., 1981. – xv+412 p. 3. P.C. Eklof, A.H. Mekler, Almost free modules. Settheoretic methods. – Revised edition. NorthHolland Mathematical Library, 65, 2002. 4. F.Q. Gouvea, padic numbers, an introduction. – Springer, Universitext, 1993. – vi+282 p. MR1251959, 2nd ed. 1997, vi+298 pp. 5. P.J. Higgins, Introduction to topological groups. – London Mathematical Society Lecture Note Series, No. 15. Cambridge University Press, LondonNew York, 1974. – v+109 p. 6. I. Protasov, E. Zelenyuk, Topologies on groups determined by sequences. – Mathematical Studies Monograph Series, 4, VNTL Publishers, L’viv, 1999. – 111 p. 7. Ye.G. Zelenyuk, Ultrafilters and topologies on groups. – de Gruyter Expositions in Mathematics, V.50, 2011. – viii+219 p. 
Pages 
115128

Volume 
43

Issue 
1

Year 
2015

Journal 
Matematychni Studii

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