A homeomorphism on $s$ not conjugate to an extendable homeomorphism
HTML articles powered by AMS MathViewer
- by Jan van Mill PDF
- Proc. Amer. Math. Soc. 105 (1989), 250-253 Request permission
Abstract:
Consider $s = \Pi _{i = 1}^\infty {( - 1,1)_i}$ and its compactification $Q = \Pi _{i = 1}^\infty {\left [ { - 1,1} \right ]_i}$. Anderson and Bing asked whether for every homeomorphism $f:s \to s$ there is a homeomorphism $\phi :s \to s$ such that ${\phi ^{ - 1}}f\phi$ is extendable to a homeomorphism $\overline {{\phi ^{ - 1}}f\phi :\;} Q \to Q$. The aim of this note is to construct a counterexample to this question.References
- R. D. Anderson and R. H. Bing, A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 74 (1968), 771–792. MR 230284, DOI 10.1090/S0002-9904-1968-12044-0
- Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0478168
- T. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399–426. MR 283828, DOI 10.1090/S0002-9947-1971-0283828-7
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
- Katsuro Sakai and Raymond Y. Wong, Conjugating homeomorphisms to uniform homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), no. 1, 337–356. MR 974780, DOI 10.1090/S0002-9947-1989-0974780-0
- H. Toruńczyk, Absolute retracts as factors of normed linear spaces, Fund. Math. 86 (1974), 53–67. MR 365471, DOI 10.4064/fm-86-1-53-67
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 105 (1989), 250-253
- MSC: Primary 54H15; Secondary 57S05, 58B05
- DOI: https://doi.org/10.1090/S0002-9939-1989-0931739-2
- MathSciNet review: 931739