Hart, Klaas P.; van Mill, Jan Homogeneity and rigidity in Erdős spaces. (English) Zbl 1424.54070 Commentat. Math. Univ. Carol. 59, No. 4, 495-501 (2018). Summary: The classical Erdős 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. MSC: 54F99 Special properties of topological spaces 46A45 Sequence spaces (including Köthe sequence spaces) 54B05 Subspaces in general topology 54D65 Separability of topological spaces 54E50 Complete metric spaces 54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites Keywords:Erdős space; homogeneity; rigidity; sphere PDFBibTeX XMLCite \textit{K. P. Hart} and \textit{J. van Mill}, Commentat. Math. Univ. Carol. 59, No. 4, 495--501 (2018; Zbl 1424.54070) Full Text: DOI arXiv References: [1] Dijkstra J. J.; van Mill J., Erdös space and homeomorphism groups of manifolds, Mem. Amer. Math. Soc. 208 (2010), no. 979, 62 pages · Zbl 1204.57041 [2] van Douwen E. K., A compact space with a measure that knows which sets are homeomorphic, Adv. in Math. 52 (1984), no. 1, 1-33 · Zbl 0535.43001 · doi:10.1016/0001-8708(84)90049-5 [3] Engelking R., General Topology, Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989 · Zbl 0684.54001 [4] Erdös P., The dimension of the rational points in Hilbert space, Ann. of Math. (2) 41 (1940), 734-736 · Zbl 0025.18701 · doi:10.2307/1968851 [5] Lavrentieff, M. A., Contribution à la théorie des ensembles homéomorphes, Fund. Math. 6 (1924), 149-160 (French) · JFM 50.0143.04 · doi:10.4064/fm-6-1-149-160 [6] Lawrence L. B., Homogeneity in powers of subspaces of the real line, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3055-3064 · Zbl 0906.54008 [7] Sierpiński W., Sur un problème concernant les types de dimensions, Fund. Math. 19 (1932), 65-71 (French) · Zbl 0005.19702 · doi:10.4064/fm-19-1-65-71 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.