Extensions of topological groups do not preserve countable compactness
by
Mikhail Tkachenko
Universidad Autónoma Metropolitana
Coauthors: M. Bruguera (Universidad Politécnica de Cataluña)

Let P be a (topological or topological-algebraic) property. We say that P is a three-space property if the following holds: whenever N is a closed normal subgroup of a topological group G and both N and G/N have P, the group G also has P. It is known that compactness, completeness, precompactness, pseudocompactness, connectedness and metrizability are three-space properties in the class of topological groups. The problem of whether countable compactness is a three-space property, in the special case of products of two countably compact groups, was considered by van Douwen [2], Hart and van Mill [3], Tomita [4], among others. However, all known constructions of counterexamples make use of extra set-theoretic assumptions such as CH or MA.

Here we present a short construction which shows that countable compactness strongly fails to be a three-space property in ZFC only.

A topological group G is called sequentially complete if no sequence in G converges to a point of [G\tilde]\G, where [G\tilde] is the Raikov completion of G. Clearly, countably compact groups are sequentially complete. A group G is \omega-bounded if every countable subset of G is contained in a compact subgroup. Obviously, every \omega-bounded group is countably compact. The example below also shows that extensions of topological groups preserve neither sequential completeness nor \omega-boundedness, thus answering Question 6.1 of [1] in the negative.

Example. There exist an Abelian pseudocompact group G and a closed subgroup N of G such that N is \omega-bounded, G/N is compact and metrizable, but G is not sequentially complete. In particular, the group G is not countably compact.

[1] D. Dikranjan and M. Tkachenko, Sequentially complete groups: dimension and minimality, J. Pure Appl. Algebra 157 (2001), 215-239.

[2] E.K. van Douwen, The product of two countably compact topological groups, Trans. Amer. Math. Soc. 262 (1980), 417-427.

[3] K.P. Hart and J. van Mill, A countably compact group H such that H×H is not countably compact, Trans. Amer. Math. Soc. 323 (1991), 811-821.

[4] A.H. Tomita, A group under MA_countable whose square is countably compact but whose cube is not, Topology Appl. 91 (1999), 91-104.

Date received: May 23, 2002