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Title: Homogeneity and rigidity in Erdös spaces (English)
Author: Hart, Klaas P.
Author: van Mill, Jan
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 59
Issue: 4
Year: 2018
Pages: 495-501
Summary lang: English
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Category: math
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Summary: The classical Erdös spaces are obtained as the subspaces of real separable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively. One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different coordinates it is possible to create a rigid subspace. (English)
Keyword: Erdös space
Keyword: homogeneity
Keyword: rigidity
Keyword: sphere
MSC: 46A45
MSC: 54B05
MSC: 54D65
MSC: 54E50
MSC: 54F50
MSC: 54F99
idZBL: Zbl 06997365
idMR: MR3914715
DOI: 10.14712/1213-7243.2015.265
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Date available: 2018-12-28T15:12:05Z
Last updated: 2019-05-22
Stable URL: http://hdl.handle.net/10338.dmlcz/147553
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Reference: [4] Erdös P.: The dimension of the rational points in Hilbert space.Ann. of Math. (2) 41 (1940), 734–736. MR 0003191, 10.2307/1968851
Reference: [5] Lavrentieff, M. A.: Contribution à la théorie des ensembles homéomorphes.Fund. Math. 6 (1924), 149–160 (French). 10.4064/fm-6-1-149-160
Reference: [6] Lawrence L. B.: Homogeneity in powers of subspaces of the real line.Trans. Amer. Math. Soc. 350 (1998), no. 8, 3055–3064. MR 1458308
Reference: [7] Sierpiński W.: Sur un problème concernant les types de dimensions.Fund. Math. 19 (1932), 65–71 (French). 10.4064/fm-19-1-65-71
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