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\(\beta X\) and fixed-point free maps. (English) Zbl 0792.54037
It is proved that whenever \(f\) is a fixed-point free map of an \(n\)- dimensional paracompact space \(X\) into itself and there exists an integer \(k\) such that the size of the preimage of every point is not greater than \(k\), then there exists an open cover \(\mathcal P\) of \(X\) such that the cardinality of \(P\) is not greater than \((n+1)\cdot (k+1)\) and \(f(U) \cap U = \emptyset\) for every \(U \in {\mathcal P}\). In particular it follows that if \(f\) is a fixed-point free homeomorphism of a finite dimensional paracompact space onto itself, then the extension of \(f\) over the Čech-Stone compactification is also fixed-point free. In the 0- dimensional case the result was obtained independently by the reviewer and Dok Yong Kim [Commentat. Math. Univ. Carol. 29, No. 4, 657-663 (1988; Zbl 0687.54011)].

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
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References:
[1] Dugundji, J., Topology, (1970), Allyn and Bacon Boston, MA
[2] Engelking, R., General topology, (1989), Heldermann Berlin · Zbl 0684.54001
[3] A. Krawczyk and J. Steprāns, Fixed points, in preparation.
[4] S. Watson, Fixed points arising only in the growth of first countable spaces, Preprint. · Zbl 0838.54026
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