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Topological Group
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The three space problem in topological groups
The three space problem in topological groups,10.1016/j.topol.2005.05.009,Topology and Its Applications,Montserrat Bruguera,Mikhail Tkachenko
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The three space problem in topological groups
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Montserrat Bruguera
,
Mikhail Tkachenko
We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every
topological group
G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p=c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μspaces: one can find a pseudocompact, noncompact Abelian
topological group
G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups.
Journal:
Topology and Its Applications  TOPOL APPL
, vol. 153, no. 13, pp. 22782302, 2006
DOI:
10.1016/j.topol.2005.05.009
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Citations
(2)
Bounded sets in topological groups and embeddings
Montserrat Bruguera
,
Mikhail Tkachenko
Journal:
Topology and Its Applications  TOPOL APPL
, vol. 154, no. 7, pp. 12981306, 2007
Supersequences and bounded sets in topological groups. Especial embeddings
Mikhail Tkachenko
Published in 2000.