| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2018-12-28T15:12:05Z |
| Last updated:
|
2021-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147553 |
| . |
| Reference:
|
[1] Dijkstra J. J., van Mill J.: Erdös space and homeomorphism groups of manifolds.Mem. Amer. Math. Soc. 208 (2010), no. 979, 62 pages. MR 2742005 |
| Reference:
|
[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. MR 0742164, 10.1016/0001-8708(84)90049-5 |
| Reference:
|
[3] Engelking R.: General Topology.Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. Zbl 0684.54001, MR 1039321 |
| 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 |
| . |
