Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
Stone-Čech compactification; right cancellable ultrafilters; left invariant topologies
Stone-Čech compactification; right cancellable ultrafilters; left invariant topologies
Summary:
For every discrete group $G$, the Stone-Čech compactification $\beta G$ of $G$ has a natural structure of a compact right topological semigroup. An ultrafilter $p\in G^*$, where $G^*=\beta G\setminus G$, is called right cancellable if, given any $q,r\in G^*$, $qp=rp$ implies $q=r$. For every right cancellable ultrafilter $p\in G^*$, we denote by $G(p)$ the group $G$ endowed with the strongest left invariant topology in which $p$ converges to the identity of $G$. For any countable group $G$ and any right cancellable ultrafilters $p,q\in G^*$, we show that $G(p)$ is homeomorphic to $G(q)$ if and only if $p$ and $q$ are of the same type.
References:
[1] Hindman N., Strauss D.: Algebra in the Stone-Čech Compactification: Theory and Applications. Walter de Gruyter, Berlin, 1998. MR 1642231 | Zbl 0918.22001
[2] Hindman N., Protasov I., Strauss D.: Topologies on $S$ determined by idempotents in $\beta S$. Topology Proc. 23 (1998), 155--190. MR 1803247 | Zbl 0970.54036
[3] Protasov I.V.: Maximal topologies on groups. Siberian Math. J. 39 (1998), 1184--1194. DOI 10.1007/BF02674129 | MR 1672661 | Zbl 0935.22002
[4] Protasov I.V.: Extremal topologies on groups. Mat. Stud. 15 (2001), 9--22. MR 1871923 | Zbl 0989.22003
[5] Protasov I.V.: Remarks on extremally disconnected semitopological groups. Comment. Math. Univ. Carolin. 43 (2002), 343--347. MR 1922132 | Zbl 1090.54033
[6] Vaughan J.E.: Two spaces homeomorphic to $Seq(p)$. Comment. Math. Univ. Carolin. 42 (2001), 209--218. MR 1825385 | Zbl 1053.54033
Search
Partner of
