On the LC1-spaces which are Cantor or arcwise homogeneous

Hanna Patkowska

Fundamenta Mathematicae (1993)

  • Volume: 142, Issue: 2, page 139-146
  • ISSN: 0016-2736

Abstract

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A space X containing a Cantor set (an arc) is Cantor (arcwise) homogeneousiff for any two Cantor sets (arcs) A,B ⊂ X there is an autohomeomorphism h of X such that h(A)=B. It is proved that a continuum (an arcwise connected continuum) X such that either dim X=1 or X ∈ L C 1 is Cantor (arcwise) homogeneous iff X is a closed manifold of dimension at most 2.

How to cite

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Patkowska, Hanna. "On the LC1-spaces which are Cantor or arcwise homogeneous." Fundamenta Mathematicae 142.2 (1993): 139-146. <http://eudml.org/doc/211977>.

@article{Patkowska1993,
abstract = {A space X containing a Cantor set (an arc) is Cantor (arcwise) homogeneousiff for any two Cantor sets (arcs) A,B ⊂ X there is an autohomeomorphism h of X such that h(A)=B. It is proved that a continuum (an arcwise connected continuum) X such that either dim X=1 or $X ∈ LC^1$ is Cantor (arcwise) homogeneous iff X is a closed manifold of dimension at most 2. },
author = {Patkowska, Hanna},
journal = {Fundamenta Mathematicae},
keywords = {Cantor homogeneous; arcwise homogeneous},
language = {eng},
number = {2},
pages = {139-146},
title = {On the LC1-spaces which are Cantor or arcwise homogeneous},
url = {http://eudml.org/doc/211977},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Patkowska, Hanna
TI - On the LC1-spaces which are Cantor or arcwise homogeneous
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 2
SP - 139
EP - 146
AB - A space X containing a Cantor set (an arc) is Cantor (arcwise) homogeneousiff for any two Cantor sets (arcs) A,B ⊂ X there is an autohomeomorphism h of X such that h(A)=B. It is proved that a continuum (an arcwise connected continuum) X such that either dim X=1 or $X ∈ LC^1$ is Cantor (arcwise) homogeneous iff X is a closed manifold of dimension at most 2.
LA - eng
KW - Cantor homogeneous; arcwise homogeneous
UR - http://eudml.org/doc/211977
ER -

References

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  1. [A] R. D. Anderson, A characterization of the universal curve and a proof of its homogeneity, Ann. of Math. 67 (1958), 313-324. Zbl0083.17607
  2. [E] E. G. Effros, Transformation groups and C ∞ -algebras, ibid. 81 (1965), 38-55. 
  3. [Hu] S.-T. Hu, Theory of Retracts, Wayne St. Univ. Press, Detroit 1965. Zbl0145.43003
  4. [L-W] J. Lamoreaux and D. G. Wright, Rigid sets in the Hilbert cube, Topology Appl. 22 (1986), 85-96. Zbl0587.54022
  5. [M] J. van Mill, Infinite-Dimensional Topology, North-Holland, 1989. Zbl0663.57001
  6. [O-P] K. Omiljanowski and H. Patkowska, On the continua which are Cantor homogeneous or arcwise homogeneous, Colloq. Math. 58 (1990), 201-212. Zbl0712.54024
  7. [U] G. S. Ungar, On all kinds of homogeneous spaces, Trans. Amer. Math. Soc. 212 (1975), 393-400. Zbl0318.54037
  8. [Wh1] G. T. Whyburn, Local separating points of continua, Monatsh. Math. Phys. 36 (1929), 305-314. Zbl55.0320.01
  9. [Wh2] G. T. Whyburn, Concerning the proposition that every closed, compact and totally disconnected set of points is a subset of an arc, Fund. Math. 18 (1932), 47-60. Zbl58.0643.03
  10. [Y] G. S. Young, Characterizations of 2-manifolds, Duke Math. J. 14 (1947), 979-990. Zbl0029.23204

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