Download PDF - On homogeneous totally disconnected 1-dimensional spacesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

On homogeneous totally disconnected 1-dimensional spaces

Authors 1, 2, 3, 1

Affiliations

  1. University of Saskatchewan, Saskatoon, Canada S7N OWO
  2. Institute of Mathematics, University of Tsukuba, Tsukuba-shi, Ibaraki 305, Japan
  3. University of Alabama at Birmingham, Birmingham, Alabama 35294, U.S.A.

Abstract

The Cantor set and the set of irrational numbers are examples of 0-dimensional, totally disconnected, homogeneous spaces which admit elegant characterizations and which play a crucial role in analysis and dynamical systems. In this paper we will start the study of 1-dimensional, totally disconnected, homogeneous spaces. We will provide a characterization of such spaces and use it to show that many examples of such spaces which exist in the literature in various fields are all homeomorphic. In particular, we will show that the set of endpoints of the universal separable ℝ-tree, the set of endpoints of the Julia set of the exponential map, the set of points in Hilbert space all of whose coordinates are irrational and the set of endpoints of the Lelek fan are all homeomorphic. Moreover, we show that these spaces satisfy a topological scaling property: all non-empty open subsets and all complements of σ-compact subsets are homeomorphic.

Keywords

ENG
totally disconnected, homogeneous, complete

Bibliography

  1. J. M. Aarts and L. G. Oversteegen, The geometry of Julia sets in the exponential family, Trans. Amer. Math. Soc. 338 (1993), 897-918.
  2. P. Alexandroff und P. Urysohn, Über nulldimensionale Punktmengen, Math. Ann. 98 (1928), 89-106.
  3. R. Bennett, Countable dense homogeneous spaces, Fund. Math. 74 (1972), 189-194.
  4. L. E. J. Brouwer, On the structure of perfect sets of points, Proc. Acad. Amsterdam 12 (1910), 785-794.
  5. W. T. Bula and L. G. Oversteegen, A characterization of smooth Cantor bouquets, Proc. Amer. Math. Soc. 108 (1990), 529-534.
  6. W. J. Charatonik, The Lelek fan is unique, Houston J. Math. 15 (1989), 27-34.
  7. F. van Engelen, Homogeneous Borel sets of ambiguous class two, Trans. Amer. Math. Soc. 290 (1985), 1-39.
  8. F. van Engelen, Homogeneous zero-dimensional absolute Borel sets, PhD thesis, Universiteit van Amsterdam, Amsterdam, 1985.
  9. P. Erdős, The dimension of rational points in Hilbert space, Ann. of Math. 41 (1940), 734-736.
  10. K. Kawamura, L. Oversteegen and E. D. Tymchatyn, On the set of endpoints of the Lelek fan, in preparation.
  11. B. Knaster, Sur les coupures biconnexes des espaces euclidiens de dimension n > 1 arbitraire, Mat. Sb. 19 (1946), 9-18 (in: Russian; French summary).
  12. K. Kuratowski et B. Knaster, Sur les ensembles connexes, Fund. Math. 2 (1921), 206-255.
  13. A. Lelek, On plane dendroids and their endpoints in the classical sense, Fund. Math. 49 (1961), 301-319.
  14. J. C. Mayer, An explosion point for the set of endpoints of the Julia set of λ exp(z), Ergodic Theory Dynam. Systems 10 (1990), 177-183.
  15. J. C. Mayer, L. Mohler, L. G. Oversteegen and E. D. Tymchatyn, Characterization of separable metric ℝ-trees, Proc. Amer. Math. Soc. 115 (1992), 257-264.
  16. J. C. Mayer, J. Nikiel and L. G. Oversteegen, On universal ℝ-trees, Trans. Amer. Math. Soc. 334 (1992), 411-432.
  17. J. C. Mayer and L. G. Oversteegen, A topological characterization of ℝ-trees, Trans. Amer. Math. Soc. 320 (1990), 395-415.
  18. S. Mazurkiewicz, Sur les problèmes χ et λ de Urysohn, Fund. Math. 10 (1927), 311-319.
  19. J. van Mill, Characterization of some zero-dimensional separable metric spaces, Trans. Amer. Math. Soc. 264 (1981), 205-215.
  20. T. Nishiura and E. D. Tymchatyn, Hereditarily locally connected spaces, Houston J. Math. 2 (1976), 581-599.
  21. L. G. Oversteegen and E. D. Tymchatyn, On the dimension of some totally disconnected sets, Proc. Amer. Math. Soc., to appear.
  22. J. H. Roberts, The rational points in Hilbert space, Duke Math. J. 23 (1956), 488-491.
  23. L. R. Rubin, Totally disconnected spaces and infinite cohomological dimension, Topology Proc. 7 (1982), 157-166.
  24. L. R. Rubin, R. M. Schori and J. J. Walsh, New dimension-theory techniques for constructing infinite dimensional examples, Gen. Topology Appl. 10 (1979), 93-102.
  25. W. Sierpiński, Sur les ensembles connexes et non connexes, Fund. Math. 2 (1921), 81-95.
  26. W. Sierpiński, Sur une propriété des ensembles dénombrables denses en soi, Fund. Math. 1 (1920), 11-16.

Additional information

http://matwbn.icm.edu.pl/ksiazki/fm/fm150/fm15021.pdf

Fundamenta Mathematicae
1996/150/2
Export to PBN, Opens in new tab
Pages:
97-112
Main language of publication
English
Received
1993-05-14
Accepted
1993-11-25
Published
1996
Exact and natural sciences