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- W2011960637 abstract "In this paper the preservation of the fixed point property under UV decompositions is studied. It is shown that if K is an n-dimensional complex with the fixed point property and G is UVn-1 decomposition of K, then KIG also will have the fixed point property. Furthermore, if X is a compact metric space with the fixed point property, and G is a UVn decomposition of Xsuch that X/G may be embedded in a suitably small Euclidian space, Rm, then X/G retains the fixed point property. Introduction. A space X is said to have the fixed point property if each continuous function f: X-+X leaves some point of X fixed, i.e., there is a point xeX such thatf(x)=x. In this paper it is shown that the fixed point property for compact metric spaces is preserved under suitable UV decompositions. Examples and some applications of the main results are described. Proofs rely to a great extent on techniques utilized by Armentrout and Price in [1], and it would be helpful if the reader is familiar with this paper. Notation and terminology. If G is an upper semicontinuous decomposition of a topological space X, then X/G will denote the associated decomposition space, and P: X-+X/G the natural projection from X onto X/G. Suppose X is a topological space, M is a subset of X, and n is a nonnegative integer. M has property nUV if and only if for each open set U containing M, there is an open set V containing M such that (1) Vc U and (2) each singular n-sphere in V is homotopic to 0 in U. M has property UV' if and only if for each nonnegative integer such that i<n, M has property i-UV; M has property UV' if for each nonnegative integer k, M has property k-UV. If X is a topological space and n is a nonnegative integer, the statement that G is UVn decomposition of X means that G is an upper semicontinuous decomposition of X into compact sets, each with property UVn. Received by the editors April 15, 1971. AMS 1969 subject classiflcations. Primary 54B1 5; Secondary 54B25. Key w,ords andphrases. Simplicial complex, compact metric space, UVdecomposition, fixed point property. @ American Mathematical Society 1972" @default.
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- W2011960637 date "1972-01-01" @default.
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- W2011960637 title "Some fixed point results for $UV$ decompositions of compact metric spaces" @default.
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- W2011960637 doi "https://doi.org/10.1090/s0002-9939-1972-0290340-4" @default.
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