Open subspaces of countable dense homogeneous spaces

Stephen Watson; Petr Simon

Fundamenta Mathematicae (1992)

  • Volume: 141, Issue: 2, page 101-108
  • ISSN: 0016-2736

Abstract

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We construct a completely regular space which is connected, locally connected and countable dense homogeneous but not strongly locally homogeneous. The space has an open subset which has a unique cut-point. We use the construction of a C 1 -diffeomorphism of the plane which takes one countable dense set to another.

How to cite

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Watson, Stephen, and Simon, Petr. "Open subspaces of countable dense homogeneous spaces." Fundamenta Mathematicae 141.2 (1992): 101-108. <http://eudml.org/doc/211955>.

@article{Watson1992,
abstract = {We construct a completely regular space which is connected, locally connected and countable dense homogeneous but not strongly locally homogeneous. The space has an open subset which has a unique cut-point. We use the construction of a $C^1$-diffeomorphism of the plane which takes one countable dense set to another.},
author = {Watson, Stephen, Simon, Petr},
journal = {Fundamenta Mathematicae},
language = {eng},
number = {2},
pages = {101-108},
title = {Open subspaces of countable dense homogeneous spaces},
url = {http://eudml.org/doc/211955},
volume = {141},
year = {1992},
}

TY - JOUR
AU - Watson, Stephen
AU - Simon, Petr
TI - Open subspaces of countable dense homogeneous spaces
JO - Fundamenta Mathematicae
PY - 1992
VL - 141
IS - 2
SP - 101
EP - 108
AB - We construct a completely regular space which is connected, locally connected and countable dense homogeneous but not strongly locally homogeneous. The space has an open subset which has a unique cut-point. We use the construction of a $C^1$-diffeomorphism of the plane which takes one countable dense set to another.
LA - eng
UR - http://eudml.org/doc/211955
ER -

References

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  1. [1] L. V. Ahlfors, Complex Analysis, McGraw-Hill, New York 1953. Zbl0052.07002
  2. [2] R. D. Anderson, D. W. Curtis and J. van Mill, A fake topological Hilbert space, Trans. Amer. Math. Soc. 272 (1982), 311-321. 
  3. [3] J. W. Bales, Representable and strongly locally homogeneous spaces and strongly n-homogeneous spaces, Houston J. Math. 2 (1976), 315-327. Zbl0341.54047
  4. [4] K. F. Barth and W. J. Schneider, Entire functions mapping arbitrary countable dense sets and their complements onto each other, J. London Math. Soc. 4 (1971/72), 482-488. Zbl0228.30020
  5. [5] K. F. Barth and W. J. Schneider, Entire functions mapping countable dense subsets of the reals onto each other monotonically, ibid. 2 (1970), 620-626. Zbl0201.09203
  6. [6] R. Bennett, Countable dense homogeneous spaces, Fund. Math. 74 (1972), 189-194. Zbl0227.54020
  7. [7] R. B. Burckel and S. Saeki, Additive mappings on rings of holomorphic functions, Proc. Amer. Math. Soc. 89 (1983), 79-85. Zbl0532.30040
  8. [8] G. Cantor, Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann. 46 (1895), 481-512. Zbl26.0081.01
  9. [9] J. de Groot, Topological Hilbert space and the drop-out effect, Technical Report zw-1969, Math. Centrum, Amsterdam 1969. Zbl0199.25702
  10. [10] T. Dobrowolski, On smooth countable dense homogeneity, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 627-634. Zbl0332.58005
  11. [11] C. Eberhart, Some remarks on the irrational and rational numbers, Amer. Math. Monthly 84 (1977), 32-35. Zbl0358.54019
  12. [12] P. Erdős, Problem 2.31, in: Research Problems in Function Theory, W. K. Haymann (ed.), Athlone Press, London 1967. 
  13. [13] P. Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), 291-300. 
  14. [14] B. Fitzpatrick, Jr., A note on countable dense homogeneity, Fund. Math. 75 (1972), 33-34. 
  15. [15] B. Fitzpatrick, Jr., and H.-X. Zhou, Densely homogeneous spaces (II), Houston J. Math. 14 (1988), 57-68. Zbl0677.54014
  16. [16] B. Fitzpatrick, Some open problems in densely homogeneous spaces, in: Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, Amsterdam 1990, 251-259. 
  17. [17] B. Fitzpatrick, Jr., and N. F. Lauer, Densely homogeneous spaces (I), Houston J. Math. 13 (1987), 19-25. Zbl0635.54009
  18. [18] P. Fletcher and R. A. McCoy, Conditions under which a connected representable space is locally connected, Pacific J. Math. 51 (1974), 433-437. Zbl0293.54013
  19. [19] L. R. Ford, Jr., Homeomorphism groups and coset spaces, Trans. Amer. Math. Soc. 77 (1954), 490-497. Zbl0058.17302
  20. [20] M. K. Fort, Jr., Homogeneity of infinite products of manifolds with boundary, Pacific J. Math. 12 (1962), 879-884. Zbl0112.38004
  21. [21] P. Franklin, Analytic transformations of everywhere dense point sets, Trans. Amer. Math. Soc. 27 (1925), 91-100. Zbl51.0166.01
  22. [22] F. Gross, Research problem 19, Bull. Amer. Math. Soc. 71 (1965), 853. 
  23. [23] F. D. Hammer and W. Knight, Problem and solution 5955, Amer. Math. Monthly 82 (1975), 415-416. 
  24. [24] K. Kuperberg, W. Kuperberg and W. R. R. Transue, On the 2-homogeneity of Cartesian products, Fund. Math. 110 (1980), 131-134. Zbl0475.54025
  25. [25] W. D. Maurer, Conformal equivalence of countable dense sets, Proc. Amer. Math. Soc. 18 (1967), 269-270. Zbl0189.36204
  26. [26] Z. A. Melzak, Existence of certain analytic homeomorphisms, Canad. Math. Bull. 2 (1959), 71-75. Zbl0089.27603
  27. [27] M. Morayne, Measure preserving analytic diffeomorphisms of countable dense sets in C n and ℝ n , Colloq. Math. 52 (1987), 93-98. Zbl0627.28012
  28. [28] B. H. Neumann and R. Rado, Monotone functions mapping the set of rational numbers onto itself, J. Austral. Math. Soc. 3 (1963), 282-287. Zbl0136.35103
  29. [29] J. W. Nienhuys and J. G. F. Thiemann, On the existence of entire functions mapping countable dense sets onto each other, Indag. Math. 38 (1976), 331-334. Zbl0336.30009
  30. [30] D. Ravdin, Various types of local homogeneity, Pacific J. Math. 50 (1974), 589-594. Zbl0295.54048
  31. [31] W. L. Saltsman, Some homogeneity problems in point set theory, Ph.D. thesis, Auburn University, 1989. 
  32. [32] D. Sato and S. Rankin, Entire functions mapping countable dense subsets of the reals onto each other monotonically, Bull. Austral. Math. Soc. 10 (1974), 67-70. Zbl0275.30020
  33. [33] P. Stäckel, Ueber arithmetische Eigenschaften analytischer Functionen, Math. Ann. 46 (1895), 513-520. 
  34. [34] J. Steprāns and S. Watson, Homeomorphisms of manifolds with prescribed behaviour on large dense sets, Bull. London Math. Soc. 19 (1987), 305-310. Zbl0637.03051
  35. [35] J. Steprāns and H. X. Zhou, Some results on CDH spaces - I, Topology Appl. 28 (1988), 147-154. Zbl0647.54017
  36. [36] E. Strauss, Problem 6, in: Entire Functions and Related Parts of Analysis, J. Korevaar (ed.), Amer. Math. Soc., Providence, R.I., 1968, 534. 
  37. [37] G. S. Ungar, Countable dense homogeneity and n-homogeneity, Fund. Math. 99 (1978), 155-160. Zbl0392.54017
  38. [38] J. van Mill, Strong local homogeneity does not imply countable dense homogeneity, Proc. Amer. Math. Soc. 84 (1982), 143-148. Zbl0486.54012
  39. [39] S. Watson, The homogeneity of small manifolds, Topology Proc. 13 (1990), 365-370. Zbl0697.57008

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