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A rigid space $ X$ for which $ X\times X$ is homogeneous; an application of infinite-dimensional topology

Author: Jan van Mill
Journal: Proc. Amer. Math. Soc. 83 (1981), 597-600
MSC: Primary 54G15; Secondary 57N20
MathSciNet review: 627701
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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.

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  • [1] A. V. Arhangel'skiĭ, Structure and classification of topological spaces and cardinal invariants, Russian Math. Surveys 33 (1978), 33-96. MR 526012 (80i:54005)
  • [2] T. A. Chapman, Lectures on Hilbert cube manifolds, CBMS Regional Conf. Series in Math., no. 28, Amer. Math. Soc., Providence, R. I., 1976. MR 0423357 (54:11336)
  • [3] R. J. Daverman, Embedding phenomena based upon decomposition theory: Wild Cantor sets satisfying strong homogeneity properties, Proc. Amer. Math. Soc. 75 (1979), 177-182. MR 529237 (80k:57031)
  • [4] -, A strongly homogeneous but wildly embedded Cantor set in the Hilbert cube (to appear).
  • [5] W. E. Haver, Mappings between ANR's that are fine homotopy equivalences, Pacific J. Math. 58 (1975), 457-461. MR 0385865 (52:6724)
  • [6] G. Kozlowski, Images of ANR's, Trans. Amer. Math. Soc. (to appear).
  • [7] K. Kuratowski, Sur la puissance de l'ensembele des "nombres de dimension" de M. Fréchet, Fund. Math. 8 (1925), 201-208.
  • [8] A. G. Kurosh, The theory of groups. II, Chelsea, New York, 1960. MR 0109842 (22:727)
  • [9] J. van Mill, Homogeneous subsets of the real line (to appear). MR 660152 (83h:54048)
  • [10] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [11] H. Toruńczyk, On CE-images of the Hilbert cube and characterization of $ Q$-manifolds, Fund. Math. 106 (1980), 31-40. MR 585543 (83g:57006)
  • [12] R. Y. T. Wong, A wild Cantor set in the Hilbert cube, Pacific J. Math. 24 (1968), 189-193. MR 0221487 (36:4539)

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Article copyright: © Copyright 1981 American Mathematical Society

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