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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2009, Issue 1, Pages 83–110 (Mi adm110)

RESEARCH ARTICLE

Algebra in the Stone-$\check C$ech compactification: applications to topologies on groups

I. V. Protasov

Department of Cybernetics, Kyiv National University, Volodimirska 64, Kyiv 01033, Ukraine

Abstract: For every discrete group $G$, the Stone-$\check{C}$ech compactification $\beta G$ of $G$ has a natural structure of compact right topological semigroup. Assume that $G$ is endowed with some left invariant topology $\Im$ and let $\overline{\tau}$ be the set of all ultrafilters on $G$ converging to the unit of $G$ in $\Im$. Then $\overline{\tau}$ is a closed subsemigroup of $\beta G$. We survey the results clarifying the interplays between the algebraic properties of $\overline{\tau}$ and the topological properties of $(G,\Im)$ and apply these results to solve some open problems in the topological group theory.
The paper consists of 13 sections: Filters on groups, Semigroup of ultrafilters, Ideals, Idempotents, Equations, Continuity in $\beta G$ and $G^*$, Ramsey-like ultrafilters, Maximality, Refinements, Resolvability, Potential compactness and ultraranks, Selected open questions.

Keywords: Stone-$\check{C}$ech compactification, product of ultrafilters, idempotents, ideals, maximality, resolvability, extremal disconnectedness.

Full text: PDF file (334 kB)

Bibliographic databases:
MSC: 22A05, 22A15, 22A20, 05A18, 54A35, 54D80
Revised: 02.05.2009
Language:

Citation: I. V. Protasov, “Algebra in the Stone-$\check C$ech compactification: applications to topologies on groups”, Algebra Discrete Math., 2009, no. 1, 83–110

Citation in format AMSBIB
\Bibitem{Pro09} \by I.~V.~Protasov \paper Algebra in the Stone-$\check C$ech compactification: applications to topologies on groups \jour Algebra Discrete Math. \yr 2009 \issue 1 \pages 83--110 \mathnet{http://mi.mathnet.ru/adm110} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2542497} \zmath{https://zbmath.org/?q=an:1199.22002}