Glasnik Matematicki, Vol. 32, No.1 (1997), 99-124.

SHAPE AND UNIFORM PROPERTIES OF HYPERSPACES OF NONCOMPACT SPACES

Takahisa Miyata and Jack Segal

Namazu College of Technology, 3600 OOka, Namazu 410, Japan
e-mail: miyata@cce.numazu-ct.ac.jp

Department of Mathematics, University of Washington, Seattle, WA 98195, USA
e-mail: segal@math.washington.edu


Abstract.   In this paper we study various shape and uniform properties of hyperspaces of noncompact spaces. In the first part we study shape properties of hyperspaces C(X) of nonempty compact subsets and symmetric products Fn(X)  (n in N U {infinity}) of noncompact spaces X in a systematic way, using approximate resolutions in the sense of Mardesic and Watanabe. This enables us to generalize the Kodama, Spiez and Watanabe results for the compact case, and also the Dold and Thom result on the infinite symmetric products of pointed CW-complexes. We also study approximate resolutions of Fn(X) for noncompact spaces X and give many applications. In the second part we study the uniform contractibility property of hyperspaces C(X) and 2X of nonempty closed subsets of noncompact spaces X and discuss the uniform boundedness of X.

1991 Mathematics Subject Classification.   54B20, 54C56, 54E15, 55P55, 55Q07.

Key words and phrases.   Hyperspace, symmetric product, uniform space, approximate resolution, shape, uniform contractibility, uniformly bounded, noncompact space.


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