Mathematics Department ] CSU ] CSUB ] Stiern Library ] Raczkowski ] 

Zbl 1007.22003
Raczkowski, S.U.
Totally bounded topological group topologies on the integers. (English)
[J] Topology Appl. 121, No.1-2, 63-74 (2002). [ISSN 0166-8641]

Author's abstract: ``We generalize an argument of {\it W. W. Comfort}, {\it F. J. Trigos-Arrieta} and {\it T. S. Wu} [Fundam. Math. 143, 119-136 (1993; Zbl 0812.22001)] showing that if there is a non-trivial 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 non-homeomorphic, and the spaces $\langle\bbfZ, \tau_B\rangle$ $(B\in{\cal B})$ are pairwise non-homeomorphic (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 non-trivial 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: