The three space problem in topological groups

Abstract

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. (M. Bruguera), mich@xanum.uam.mx (M. Tkachenko).

The extensions of paratopological groups

“…The following lemma was proved in [4] and [13]. For the sake of completeness we give the proof of the case of sequentially compact sets.…”

“…A topological property P is called an inverse fiber property [4] if (*) f : X → Y is a continuous and surjective mapping such that both the space Y and the fibers of f have P, then X also has P. If the conclusion in (*) holds under the additional assumption that the domain X is compact (countably compact), we say that P is an inverse fiber property for compact (countably compact) sets.…”

“…The first axiom of countability is an inverse fiber property for compact, countably compact [4,Proposition 2.8] and sequentially compact [13,Lemma 2.4] sets.…”

“…Let N be a closed invariant subgroup of a paratopological group G such that both N and the quotient paratopological group G/N have P. When can we conclude that G has P? Recall that, for topological groups, if the property P holds in this problem, then we say that P is a three space property [4]. In fact, the list of three space properties in topological groups is quite long.…”

“…It includes compactness [8], local compactness [16], pseudocompactness [5], metrizability [9], etc. In 2006, M. Bruguera and M. Tkachenko [4] studied some properties of compact, countably compact, pseudocompact, and functionally bounded sets which are preserved or destroyed when taking extensions of topological groups. Recently, the convergence phenomena in the extensions of topological groups were studied in [13].…”