On group topologies determined by families of sets

G. M. Bergman

doi:10.15330/ms.43.2.115-128

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.

	On group topologies determined by families of sets
Author	G. M. Bergman 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.43.2.115-128
Reference	1. G. M. Bergman, On monoids, 2-firs 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. Set-theoretic methods. – Revised edition. North-Holland Mathematical Library, 65, 2002. 4. F.Q. Gouvea, p-adic 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, London-New 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	115-128
Volume	43
Issue	1
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html