Abstract
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.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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van Mill, J. On the G-compactifications of the rational numbers. Monatsh Math 157, 257–266 (2009). https://doi.org/10.1007/s00605-008-0024-8
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DOI: https://doi.org/10.1007/s00605-008-0024-8