Geodesic
Combinatorial Lemma for Lebesgue–Brouwer Partition of Cube on Euclidean Space
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 150 (2008) no. 1, pp. 124-129 Cet article a éte moissonné depuis la source Math-Net.Ru

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The purpose of this article is to present a new combinatorial lemma that can be used in fixed point theorem, for example, to prove Brouwer theorem. Partition of a three-dimensional cube used by Lebesgue and Brouwer in dimension theory is taken into consideration. Explanation is given for generalization of the lemma on Euclidean $n$-space, $n>3$.
Keywords: combinatorial lemma, fixed point theorem. 
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     title = {Combinatorial {Lemma} for {Lebesgue{\textendash}Brouwer} {Partition} of {Cube} on {Euclidean} {Space}},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
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R. R. Shagidullin. Combinatorial Lemma for Lebesgue–Brouwer Partition of Cube on Euclidean Space. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 150 (2008) no. 1, pp. 124-129. http://geodesic.mathdoc.fr/item/UZKU_2008_150_1_a12/

[1] Prasolov V. V., Elementy kombinatornoi i differentsialnoi topologii, Izd-vo MTsNMO, M., 2004, 352 pp.