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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)
##### Keywords:
paracompact space; open cover; Čech-Stone compactification
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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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