Эта публикация цитируется в
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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
Аннотация:
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.
Ключевые слова:
Stone-$\check{C}$ech compactification, product of ultrafilters, idempotents, ideals, maximality, resolvability, extremal disconnectedness.
MSC: 22A05,
22A15,
22A20,
05A18,
54A35,
54D80 Поступила в редакцию: 09.04.2009
Исправленный вариант: 02.05.2009
Язык публикации: английский