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

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 $G$ .