On homogeneous totally disconnected 1-dimensional spaces

Kazuhiro Kawamura; Lex Oversteegen; E. Tymchatyn

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 2, page 97-112
  • ISSN: 0016-2736

Abstract

top
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.

How to cite

top

Kawamura, Kazuhiro, Oversteegen, Lex, and Tymchatyn, E.. "On homogeneous totally disconnected 1-dimensional spaces." Fundamenta Mathematicae 150.2 (1996): 97-112. <http://eudml.org/doc/212170>.

@article{Kawamura1996,
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.},
author = {Kawamura, Kazuhiro, Oversteegen, Lex, Tymchatyn, E.},
journal = {Fundamenta Mathematicae},
keywords = {totally disconnected; homogeneous; complete},
language = {eng},
number = {2},
pages = {97-112},
title = {On homogeneous totally disconnected 1-dimensional spaces},
url = {http://eudml.org/doc/212170},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Kawamura, Kazuhiro
AU - Oversteegen, Lex
AU - Tymchatyn, E.
TI - On homogeneous totally disconnected 1-dimensional spaces
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 2
SP - 97
EP - 112
AB - 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.
LA - eng
KW - totally disconnected; homogeneous; complete
UR - http://eudml.org/doc/212170
ER -

References

top
  1. [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. Zbl0809.54034
  2. [2] P. Alexandroff und P. Urysohn, Über nulldimensionale Punktmengen, Math. Ann. 98 (1928), 89-106. Zbl53.0559.01
  3. [3] R. Bennett, Countable dense homogeneous spaces, Fund. Math. 74 (1972), 189-194. Zbl0227.54020
  4. [4] L. E. J. Brouwer, On the structure of perfect sets of points, Proc. Acad. Amsterdam 12 (1910), 785-794. 
  5. [5] W. T. Bula and L. G. Oversteegen, A characterization of smooth Cantor bouquets, Proc. Amer. Math. Soc. 108 (1990), 529-534. Zbl0679.54034
  6. [6] W. J. Charatonik, The Lelek fan is unique, Houston J. Math. 15 (1989), 27-34. Zbl0675.54034
  7. [7] F. van Engelen, Homogeneous Borel sets of ambiguous class two, Trans. Amer. Math. Soc. 290 (1985), 1-39. Zbl0582.54023
  8. [8] F. van Engelen, Homogeneous zero-dimensional absolute Borel sets, PhD thesis, Universiteit van Amsterdam, Amsterdam, 1985. Zbl0599.54044
  9. [9] P. Erdős, The dimension of rational points in Hilbert space, Ann. of Math. 41 (1940), 734-736. Zbl0025.18701
  10. [10] K. Kawamura, L. Oversteegen and E. D. Tymchatyn, On the set of endpoints of the Lelek fan, in preparation. 
  11. [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). Zbl0061.40104
  12. [12] K. Kuratowski et B. Knaster, Sur les ensembles connexes, Fund. Math. 2 (1921), 206-255. 
  13. [13] A. Lelek, On plane dendroids and their endpoints in the classical sense, Fund. Math. 49 (1961), 301-319. Zbl0099.17701
  14. [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. [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. Zbl0754.54026
  16. [16] J. C. Mayer, J. Nikiel and L. G. Oversteegen, On universal ℝ-trees, Trans. Amer. Math. Soc. 334 (1992), 411-432. Zbl0787.54036
  17. [17] J. C. Mayer and L. G. Oversteegen, A topological characterization of ℝ-trees, Trans. Amer. Math. Soc. 320 (1990), 395-415. Zbl0729.54008
  18. [18] S. Mazurkiewicz, Sur les problèmes χ et λ de Urysohn, Fund. Math. 10 (1927), 311-319. 
  19. [19] J. van Mill, Characterization of some zero-dimensional separable metric spaces, Trans. Amer. Math. Soc. 264 (1981), 205-215. Zbl0493.54018
  20. [20] T. Nishiura and E. D. Tymchatyn, Hereditarily locally connected spaces, Houston J. Math. 2 (1976), 581-599. Zbl0341.54043
  21. [21] L. G. Oversteegen and E. D. Tymchatyn, On the dimension of some totally disconnected sets, Proc. Amer. Math. Soc., to appear. Zbl0817.54028
  22. [22] J. H. Roberts, The rational points in Hilbert space, Duke Math. J. 23 (1956), 488-491. 
  23. [23] L. R. Rubin, Totally disconnected spaces and infinite cohomological dimension, Topology Proc. 7 (1982), 157-166. Zbl0523.55003
  24. [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. Zbl0413.54042
  25. [25] W. Sierpiński, Sur les ensembles connexes et non connexes, Fund. Math. 2 (1921), 81-95. Zbl48.0208.02
  26. [26] W. Sierpiński, Sur une propriété des ensembles dénombrables denses en soi, Fund. Math. 1 (1920), 11-16. Zbl47.0175.03

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.