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ON THE HOMEOMORPHISM GROUPS OF CANTOR'S DISCONTINUUM AND THE SPACES OF RATIONAL AND IRRATIONAL NUMBERS

Published online by Cambridge University Press:  24 March 2003

MANFRED DROSTE  and
RÜDIGER GÖBEL
MANFRED DROSTE
Affiliation:
Institut für Algebra, Technische Universität Dresden, 01062 Dresden, Germanydroste@math.tu-dresden.de
RÜDIGER GÖBEL
Affiliation:
Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, GermanyR.Goebel@Uni-Essen.De
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Abstract

The paper shows that the homeomorphism groups of, respectively, Cantor's discontinuum, the rationals and the irrationals have uncountable cofinality. It is well known that the homeomorphism group of Cantor's discontinuum is isomorphic to the automorphism group Aut ${\bb B}$ of the countable, atomless boolean algebra ${\bb B}$ . So also Aut ${\bb B}$ has uncountable cofinality, which answers a question posed earlier by the first author and H. D. Macpherson. The cofinality of a group $G$ is the cardinality of the length of a shortest chain of proper subgroups terminating at $G$ .

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 34 , Issue 4 , July 2002 , pp. 474 - 478
Copyright
© The London Mathematical Society 2002

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