On a compactification of the homeomorphism group of the pseudo-arc

Kazuhiro Kawamura

Colloquium Mathematicae (1991)

  • Volume: 62, Issue: 2, page 325-330
  • ISSN: 0010-1354

Abstract

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A continuum means a compact connected metric space. For a continuum X, H(X) denotes the space of all homeomorphisms of X with the compact-open topology. It is well known that H(X) is a completely metrizable, separable topological group. J. Kennedy [8] considered a compactification of H(X) and studied its properties when X has various types of homogeneity. In this paper we are concerned with the compactification G P of the homeomorphism group of the pseudo-arc P, which is obtained by the method of Kennedy. We prove that G P is homeomorphic to the Hilbert cube. This is an easy consequence of a combination of the results of [2], Corollary 2, and [9], Theorem 1, but here we give a direct proof. The author wishes to thank the referee for pointing out the above reference [2]. We also prove that the remainder of H(P) in G P contains many Hilbert cubes. It is known that H(P) contains no nondegenerate continua ([10]).

How to cite

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Kawamura, Kazuhiro. "On a compactification of the homeomorphism group of the pseudo-arc." Colloquium Mathematicae 62.2 (1991): 325-330. <http://eudml.org/doc/210120>.

@article{Kawamura1991,
abstract = {A continuum means a compact connected metric space. For a continuum X, H(X) denotes the space of all homeomorphisms of X with the compact-open topology. It is well known that H(X) is a completely metrizable, separable topological group. J. Kennedy [8] considered a compactification of H(X) and studied its properties when X has various types of homogeneity. In this paper we are concerned with the compactification $G_P$ of the homeomorphism group of the pseudo-arc P, which is obtained by the method of Kennedy. We prove that $G_P$ is homeomorphic to the Hilbert cube. This is an easy consequence of a combination of the results of [2], Corollary 2, and [9], Theorem 1, but here we give a direct proof. The author wishes to thank the referee for pointing out the above reference [2]. We also prove that the remainder of H(P) in $G_P$ contains many Hilbert cubes. It is known that H(P) contains no nondegenerate continua ([10]).},
author = {Kawamura, Kazuhiro},
journal = {Colloquium Mathematicae},
language = {eng},
number = {2},
pages = {325-330},
title = {On a compactification of the homeomorphism group of the pseudo-arc},
url = {http://eudml.org/doc/210120},
volume = {62},
year = {1991},
}

TY - JOUR
AU - Kawamura, Kazuhiro
TI - On a compactification of the homeomorphism group of the pseudo-arc
JO - Colloquium Mathematicae
PY - 1991
VL - 62
IS - 2
SP - 325
EP - 330
AB - A continuum means a compact connected metric space. For a continuum X, H(X) denotes the space of all homeomorphisms of X with the compact-open topology. It is well known that H(X) is a completely metrizable, separable topological group. J. Kennedy [8] considered a compactification of H(X) and studied its properties when X has various types of homogeneity. In this paper we are concerned with the compactification $G_P$ of the homeomorphism group of the pseudo-arc P, which is obtained by the method of Kennedy. We prove that $G_P$ is homeomorphic to the Hilbert cube. This is an easy consequence of a combination of the results of [2], Corollary 2, and [9], Theorem 1, but here we give a direct proof. The author wishes to thank the referee for pointing out the above reference [2]. We also prove that the remainder of H(P) in $G_P$ contains many Hilbert cubes. It is known that H(P) contains no nondegenerate continua ([10]).
LA - eng
UR - http://eudml.org/doc/210120
ER -

References

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  1. [1] R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729-742. Zbl0035.39103
  2. [2] P. L. Bowers, Dense embeddings of nowhere locally compact separable metric spaces, Topology Appl. 26 (1987), 1-12. Zbl0624.54014
  3. [3] M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478-483. Zbl0113.37705
  4. [4] T. A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. 28, Amer. Math. Soc. Providence, R.I., 1975. 
  5. [5] D. W. Curtis and R. M. Schori, Hyperspaces which characterize simple homotopy type, Gen. Topology Appl. 6 (1976), 153-165. Zbl0328.54003
  6. [6] K. Kawamura, Span zero continua and the pseudo-arc, Tsukuba J. Math. 14 (1990), 327-341. Zbl0746.54013
  7. [7] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36. Zbl0061.40107
  8. [8] J. Kennedy, Compactifying the space of homeomorphisms, Colloq. Math. 56 (1988), 41-58. Zbl0676.54047
  9. [9] J. Kennedy Phelps, Homogeneity and groups of homeomorphisms, Topology Proc. 6 (1981), 371-404. Zbl0525.54025
  10. [10] W. Lewis, Pseudo-arc and connectedness in homeomorphism groups, Proc. Amer. Math. Soc. 87 (1983), 745-748. Zbl0525.54024
  11. [11] S. B. Nadler, Hyperspaces of Sets, Marcel Dekker, 1978. Zbl0432.54007
  12. [12] S. B. Nadler, Induced universal maps and some hyperspaces with fixed point property, Proc. Amer. Math. Soc. 100 (1987), 749-754. Zbl0622.54006
  13. [13] M. Smith, Concerning the homeomorphisms of the pseudo-arc X as a subspace of C(X×X), Houston J. Math. 12 (1986), 431-440. Zbl0629.54001
  14. [14] H. Toruńczyk, On CE-images of the Hilbert cube and the characterization of Q-manifolds, Fund. Math. 106 (1980), 31-40. Zbl0346.57004

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