Zbl 1007.22003
Raczkowski, S.U.
Totally bounded topological group topologies on the integers. (English)
[J]
Topology Appl. 121, No.12, 6374 (2002). [ISSN 01668641]
Author's abstract: ``We generalize an argument of {\it W. W. Comfort}, {\it F.
J. TrigosArrieta} and {\it T. S. Wu} [Fundam. Math. 143, 119136 (1993;
Zbl 0812.22001)] showing that if there is a nontrivial sequence converging
to the identity in a locally compact Abelian group $G$, then $A:=
\{\lambda\in\widehat G:\lambda(x_n)\to 1\}$ is a locally $\mu$null subgroup of
the character group $\widehat G$ of $G$, where $\mu$ denotes Haar measure on
$\widehat G$. Using a result of the same authors we show the existence of
families ${\cal A}$ and ${\cal B}$ of dense subgroups of $\bbfT\simeq
\widehat{\bbfZ}$ such that:\par (i) ${\cal A}={\cal B}= 2^{\germ c}$;\par
(ii) each $A\in{\cal A}$ and each $B\in{\cal B}$ is algebraically isomorphic to
the free Abelian group $\oplus_{\germ c}\bbfZ$;\par (iii) the spaces
$\langle\bbfZ, \tau_A\rangle$ $(A\in{\cal A})$ are pairwise nonhomeomorphic,
and the spaces $\langle\bbfZ, \tau_B\rangle$ $(B\in{\cal B})$ are pairwise
nonhomeomorphic (by $\tau_B$ we denote the weakest topology making all
elements of $X$ continuous);\par (iv) each group $\langle\bbfZ, \tau_A\rangle$
$(A\in{\cal A})$ has a nontrivial convergent sequence; and\par (v) every
convergent sequence of $\langle\bbfZ, \tau_B\rangle$ $(B\in{\cal B})$ is
trivial''.
[
T.S.Wu (Cleveland) ]
 MSC 2000:
