On the G-compactifications of the rational numbers

By Jan van Mill. 2009

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We show that for any sufficiently homogeneous metrizable compactum X there is a Polish group G acting continuously on the space of rational numbers $${\mathbb{Q}}$$ such that X is its unique G-compactification. This allows us to answer Problem 995 in the ‘Open Problems in Topology II’ book in the negative: there is a one-dimensional Polish group G acting transitively on $${\mathbb{Q}}$$ for which the Hilbert cube is its unique G-completion.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

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