Compactness type properties in extensions of topological groups
by
Montserrat Bruguera
Universitat Politècnica de Catalunya
Coauthors: Mikhail Tkachenko

Let P be a (algebraic, topological or algebraic-topological) property. We say that P is a three space property if the following holds: whenever H is a closed invariant subgroup of G and both H and G/H have P, the group G also has P. Compactness, precompactness, pseudocompactness, completeness, connectedness and metrizability are the three space properties in the class of topological groups (see [3, 4, 5]). However, having a countable network, s-compactness, being Lindelöf, countable compactness, sequential compactness, sequential completeness and w-compactness are not three space properties (for the first three properties, this follows from an example given in [6], while for the last four properties, see [1]).

In [2], we study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. 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 m-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 groups.

1. M. Bruguera and M. Tkachenko, Extensions of topological groups do not respect countable compactness, 5 pp., submitted.

2. M. Bruguera and M. Tkachenko, The three space problem in topological groups, preprint.

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

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

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

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

Date received: June 18, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calv-01.