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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

The three space problem in topological groups   (Citations: 2)
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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 μ-spaces: one can find a pseudocompact, non-compact 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. 2278-2302, 2006
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