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

Kazuhiro Kawamura

ArticleOriginal scientific text

Title

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

Authors ¹

Affiliations

Institute of Mathematics, University of Tsukuba, Tsukuba-city, Ibaraki 305, Japan

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]).

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Title

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

Affiliations

Abstract

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