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The three space problem in topological groups. (English) Zbl 1102.54039

A property $𝒫$ is said to be a shape 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 known that compactness, precompactness pseudocompactness, completeness, connectedness and metrizability are three space properties, and, on the other hand, having a countable network, $\sigma$-compactness, Lindelöfness, countable compactness, sequential compactness, sequential completeness and $\omega$-compactness are not three space properties.

In this paper, the authors study some properties of compact, countably compact, pseudocompact and functionally bounded sets which are preserved or destroyed when taking extensions of topological groups. A property $𝒫$ is a three space property for compact (countably compact) sets if all compact (countably compact) subsets of a topological group $G$ have $𝒫$ whenever $G$ contains a closed invariant subgroup $N$ such that the compact (countably compact) subsets of both groups $N$ and $G/N$ have $𝒫$. In Section 3, they show that metrizability is a three space property for compact sets and also a three space property for countably compact sets. In Section 4, they present several examples. For example, they show that under $𝔭=𝔠$, there is an Abelian topological group $G$ and a closed subgroup $N$ of $G$ such that all pseudocompact subspaces of $N$ are finite and the quotient group $G/N$ is countable, but $G$ contains a pseudocompact subspaces of uncountable character.

##### MSC:
 54H11 Topological groups (topological aspects) 22A05 Structure of general topological groups 54A20 Convergence in general topology 54G20 Counterexamples (general topology)