A property is said to be a shape three space property if, for every topological group and a closed invariant subgroup of , the fact that both groups and have implies that also has . It is known that compactness, precompactness pseudocompactness, completeness, connectedness and metrizability are three space properties, and, on the other hand, having a countable network, -compactness, Lindelöfness, countable compactness, sequential compactness, sequential completeness and -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 is a three space property for compact (countably compact) sets if all compact (countably compact) subsets of a topological group have whenever contains a closed invariant subgroup such that the compact (countably compact) subsets of both groups and have . 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 , there is an Abelian topological group and a closed subgroup of such that all pseudocompact subspaces of are finite and the quotient group is countable, but contains a pseudocompact subspaces of uncountable character.