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The Stone-Čech compactification of the rational world

Published online by Cambridge University Press:  18 May 2009

M. P. Stannett
M. P. Stannett
Affiliation:
Department of Computer Scinece, The University, Sheffield, S3 7RH
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In his paper [11], Peter Neumann considered in detail the cycle structures of elements of Aut(ℚ), the group of all homeomorphisms of the “rational world” ℚ onto itself, and further analyses of Aut(ℚ) and its subgroups have been given by Mekler [9], Bruyns [1], and Truss [13]. My interest in Aut(ℚ) stems from its utility in proving an at first sight rather startling (to a general topologist) result concerning β ℚ, the so-called Stone-Čech compactification of ℚ, namely that βℚ\ℚ is separable, and in fact contains a homogeneous countable dense subspace. (A space X is “homogeneous” provided whenever x, y ∈ X, there is some g ∈ Aut(X) with g(x) = y.) This is in sharp contrast to the spaces βℕ\ℕ and βℝ\ℝ, which are both inseparable.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 2 , May 1988 , pp. 181 - 188
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

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