A rigid space $X$ for which $X\times X$ is homogeneous; an application of infinite-dimensional topology
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- by Jan van Mill
- Proc. Amer. Math. Soc. 83 (1981), 597-600
- DOI: https://doi.org/10.1090/S0002-9939-1981-0627701-2
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Abstract:
We give an example of a rigid (= no autohomeomorphisms beyond the identity) space $X$ such that $X \times X$ is homogeneous. In fact, $X \times X$ is homeomorphic to the Hilbert cube. This answers a question of A. V. Arhangel’skiĭ.References
- A. V. Arhangel′skiĭ, The structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk 33 (1978), no. 6(204), 29–84, 272 (Russian). MR 526012
- T. A. Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Mathematics, No. 28, American Mathematical Society, Providence, R.I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975. MR 0423357
- Robert J. Daverman, Embedding phenomena based upon decomposition theory: wild Cantor sets satisfying strong homogeneity properties, Proc. Amer. Math. Soc. 75 (1979), no. 1, 177–182. MR 529237, DOI 10.1090/S0002-9939-1979-0529237-7 —, A strongly homogeneous but wildly embedded Cantor set in the Hilbert cube (to appear).
- William E. Haver, Mappings between $\textrm {ANR}$s that are fine homotopy equivalences, Pacific J. Math. 58 (1975), no. 2, 457–461. MR 385865 G. Kozlowski, Images of ANR’s, Trans. Amer. Math. Soc. (to appear). K. Kuratowski, Sur la puissance de l’ensembele des "nombres de dimension" de M. Fréchet, Fund. Math. 8 (1925), 201-208.
- A. G. Kurosh, The theory of groups, Chelsea Publishing Co., New York, 1960. Translated from the Russian and edited by K. A. Hirsch. 2nd English ed. 2 volumes. MR 0109842
- Jan van Mill, Homogeneous subsets of the real line, Compositio Math. 46 (1982), no. 1, 3–13. MR 660152
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112
- H. Toruńczyk, On $\textrm {CE}$-images of the Hilbert cube and characterization of $Q$-manifolds, Fund. Math. 106 (1980), no. 1, 31–40. MR 585543, DOI 10.4064/fm-106-1-31-40
- Raymond Y. T. Wong, A wild Cantor set in the Hilbert cube, Pacific J. Math. 24 (1968), 189–193. MR 221487
Bibliographic Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 83 (1981), 597-600
- MSC: Primary 54G15; Secondary 57N20
- DOI: https://doi.org/10.1090/S0002-9939-1981-0627701-2
- MathSciNet review: 627701