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Proceedings of the American Mathematical Society

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Endohomeomorphisms decomposing a space into disjoint copies of a subspace

Author: Liam O’Callaghan
Journal: Proc. Amer. Math. Soc. 72 (1978), 391-396
MSC: Primary 54C45; Secondary 54D35
MathSciNet review: 0500820
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Abstract: The existence (conjectured by R. Levy in a private communication) of a space X and an endohomeomorphism, f, of $ \beta X$, such that $ f[X] = \beta X\backslash X$ is demonstrated. It is shown that if G is one of the topological groups $ {{\mathbf{2}}^\alpha },{{\mathbf{Q}}^\alpha },{{\mathbf{R}}^\alpha }$ or $ {{\mathbf{T}}^\alpha }$, where $ \omega < \alpha $, then G has a dense C-embedded subgroup H and an autohomeomorphism, f, such that G is the union of disjoint sets, $ {A_0}$ and $ {A_1}$, where for $ \{ i,j\} = \{ 0,1\} f[{A_i}] = {A_j}$, and $ {A_i}$ is a union of cosets of H.

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  • [B64] I. N. Baker, Solution to advanced problem 5077 (proposed in [Kn63]), Amer. Math. Monthly 71 (1964), 219-220. MR 1532554
  • [BE51] N. G. DeBruijn and P. Erdös, A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A 54 = Indag. Math. 13 (1951), 371-373. MR 0046630 (13:763g)
  • [C77] W. Wistar Comfort, Ultrafilters: Some old and some new results, Bull. Amer. Math. Soc. 83 (1977), 417-455. MR 0454893 (56:13136)
  • [CN74] W. Wistar Comfort and S. Negrepontis, The theory of ultrafilters, Grundlehren der Math. Wiss., Band 211, Springer, Berlin and New York, 1974. MR 0396267 (53:135)
  • [CN75] -, Continuous pseudometrics, Lecture Notes in Pure and Appl. Math., Vol. 14, Marcel Dekker, New York, 1975. MR 0410618 (53:14366)
  • [CO'C79] W. W. Comfort and Liam O'Callaghan, Spaces Y homeomorphic to $ \beta Y\backslash Y$, Math. Z. (to appear). MR 512463 (80d:54026)
  • [Cn59] H. H. Corson, Normality in subsets of product spaces, Amer. J. Math. 81 (1959), 785-796. MR 0107222 (21:5947)
  • [E50] P. Erdös, Some remarks on set theory, Proc. Amer. Math. Soc. 1 (1950), 127-141. MR 0035809 (12:14c)
  • [Fk68] Z. Frolík, Fixed points of maps of extremally disconnected spaces and complete Boolean algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 269-275. MR 0233343 (38:1665)
  • [GJ60] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, N. J., 1960. (Reissued in 1976 by Springer as volume 43 of Graduate Texts in Mathematics.) MR 0116199 (22:6994)
  • [G59] I. Glicksberg, Stone-Cech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369-382. MR 0105667 (21:4405)
  • [G51] W. H. Gottschalk, Choice functions and Tychonoff's theorem, Proc. Amer. Math. Soc. 2 (1951), 172. MR 0040376 (12:683g)
  • [K67] M. Katětov, A theorem on mappings, Comment. Math. Univ. Carolinae 8 (1967), 431-433. MR 0229228 (37:4802)
  • [Kn63] H. Kenyon, Advanced problem no. 5077, Amer. Math. Monthly 70 (1963), 216.
  • [O'C76] Liam J. V. O'Callaghan, Topological endohomeomorphisms and compactness properties of products and generalized $ \Sigma $-products, Doctoral Dissertation, Wesleyan Univ., 1976.
  • [R49] R. Rado, Axiomatic treatment of rank in infinite sets, Canad. J. Math. 1 (1949), 337-343. MR 0031993 (11:238c)
  • [W74] R. C. Walker, The Stone-Cech compactification, Ergebnisse der Math. und ihrer Grenzgebiete, vol. 83, Springer-Verlag, Berlin and New York, 1974. MR 0380698 (52:1595)

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