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## Colloquium Mathematicum

1991 | 62 | 2 | 325-330
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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]).
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
325-330
Opis fizyczny
Daty
wydano
1991
otrzymano
1989-10-24
poprawiono
1990-08-30
Twórcy
autor
• Institute of Mathematics, University of Tsukuba, Tsukuba-city, Ibaraki 305, Japan
Bibliografia
• [1] R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729-742.
• [2] P. L. Bowers, Dense embeddings of nowhere locally compact separable metric spaces, Topology Appl. 26 (1987), 1-12.
• [3] M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478-483.
• [4] T. A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Regional Conf. Ser. in Math. 28, Amer. Math. Soc. Providence, R.I., 1975.
• [5] D. W. Curtis and R. M. Schori, Hyperspaces which characterize simple homotopy type, Gen. Topology Appl. 6 (1976), 153-165.
• [6] K. Kawamura, Span zero continua and the pseudo-arc, Tsukuba J. Math. 14 (1990), 327-341.
• [7] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36.
• [8] J. Kennedy, Compactifying the space of homeomorphisms, Colloq. Math. 56 (1988), 41-58.
• [9] J. Kennedy Phelps, Homogeneity and groups of homeomorphisms, Topology Proc. 6 (1981), 371-404.
• [10] W. Lewis, Pseudo-arc and connectedness in homeomorphism groups, Proc. Amer. Math. Soc. 87 (1983), 745-748.
• [11] S. B. Nadler, Hyperspaces of Sets, Marcel Dekker, 1978.
• [12] S. B. Nadler, Induced universal maps and some hyperspaces with fixed point property, Proc. Amer. Math. Soc. 100 (1987), 749-754.
• [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.
• [14] H. Toruńczyk, On CE-images of the Hilbert cube and the characterization of Q-manifolds, Fund. Math. 106 (1980), 31-40.
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Bibliografia
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