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A Dowker group. (English) Zbl 0592.22001

In this note the authors give, in ZFC, an example of a normal topological group which is not countably paracompact. This is done by applying the B(X)-construction of the first and third author [Topology Appl. 20, 279- 287 (1985; Zbl 0577.22005)] to a suitable Dowker space X. The main difficulty is to assure that the semitopological group B(X), which is a Dowker space if all finite powers of X are normal, is in fact a topological group. To this end the authors replace in M. E. Rudin’s Dowker space X the cardinals \(\omega,\omega_ 1,\omega_ 2,..\). by a sequence of successive cardinals \(\alpha,\alpha^+,..\). (\(\alpha\geq \omega)\) and show (i) B(X) is a topological group for \(\alpha \geq 2^{\omega}\) (and hence a Dowker group), (ii) B(X) is not a topological group for \(\alpha^+<2^{\omega}\); for \(\alpha^+=2^{\omega}\) the problem is unsolved.
Reviewer: V.Eberhardt

MSC:

22A05 Structure of general topological groups
54G20 Counterexamples in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

Citations:

Zbl 0577.22005
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