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Some properties of \(n\)-dimensional generalized cubes. (English) Zbl 0871.54042

The key result in the paper is a combinatorial statement similar to Sperner’s Lemma. It is used in much the same way Sperner’s Lemma is in order to prove the Brouwer Fixed Point Theorem and some related classical results that concern zeros and coincidences, as well as fixed points, of maps on \(n\)-dimensional cubes. These results are then extended to a class of spaces that are limits of certain inverse sequences of \(n\)-dimensional cubes. This class includes the pseudoarc and, as the author notes, the results of this paper not only imply that the pseudoarc has the fixed point property [O. H. Hamilton, Proc. Am. Math. Soc. 2, 173-174 (1951; Zbl 0054.07003)] but also that the cartesian product of an arbitrary family of pseudoarcs has that property.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 0054.07003
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