Compactness type properties in extensions of topological groups
by
Mikhail Tkachenko
Universidad Autónoma Metropolitana, Mexico
Coauthors: Montserrat Bruguera (Universidad Politècnica de Catalunia)

Let P be a topological property. We say that P is a three space property if the following holds: whenever N is a closed invariant subgroup of a topological group G and both N and G/N have P, the group G also has P. It is well known that compactness, pseudocompactness, precompactness, completeness, connectedness and metrizability are three space properties [2, 3, 4]. On the other hand, Uspenskij's example in [5] shows that having a countable network, sigma-compactness, Lindelöfness and omega-monolithicity are not three space properties.

We continue the study of extensions in the class of topological groups, with a special emphasis given to compact, countably compact and pseudocompact subsets of groups. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then the same conclusion remains valid for the group G. However, under CH, an analogous assertion fails to hold for pseudocompact subsets.

We construct several examples that destroy a number of tempting conjectures about extensions of topological groups. For example, it turns out that countable compactness and sequential compactness are not three space properties in ZFC only [1]. Realcompactness, Dieudonné completeness and the property of being a mu-space are not three space properties either. In the lecture, we pretend to give an updated list of such properties and describe the technique employed for constructing counterexamples.

[1] M. Bruguera and M. Tkachenko, Extensions of topological groups do not respect countable compactness, Submitted.

[2] W.W. Comfort and L. Robertson, Extremal phenomena in certain classes of totally bounded groups, Dissert. Math. 272 (1988), 1-48.

[3] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis I, Die Grundlehrender Mathematischen Wissenschaften 115 (1963).

[4] W. Roelcke and S. Dierolf, Uniform Structures on Topological Groups and their Quotients, McGraw-Hill International Book Company, New York-Toronto 1981.

[5] V.V. Uspenskij, Extensions of topological groups with a countable net, Moscow Univ. Math. Bull. 39 no. 5 (1984), 84-85.

Date received: April 22, 2003