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Parametric extension of the Poincaré theorem. (English) Zbl 0973.26010

Given a partition of the unit square \(Q= [0,1]\times [0,1]\) into two closed subsets \(A\) and \(B\) which contain the left and right border of \(Q\), respectively (i.e., \(\{0\}\times [0,1]\subseteq A\) and \(\{1\}\times [0,1]\subseteq B\)), one can always find a connected set \(C\subseteq A\cap B\) which meets both the upper and lower border of \(Q\) (i.e., \(C\cap [0,1]\times \{0\}\neq\emptyset\) and \(C\cap [0,1]\times \{1\}\neq\emptyset\)). Geometrically this is rather obvious, but a mathematically rigorous proof is not trivial at all. In this interesting paper, the authors prove this statement even in higher dimensions and deduce some consequences like a “parameter-dependent” version of Brouwer’s fixed point theorem and some related results. A collection of such results may be found in an earlier survey of the first author [Am. Math. Mon. 104, No. 6, 545-550 (1997; Zbl 0891.47040)].

MSC:

26B05 Continuity and differentiation questions
54H25 Fixed-point and coincidence theorems (topological aspects)
54C05 Continuous maps
54B15 Quotient spaces, decompositions in general topology
47H10 Fixed-point theorems

Citations:

Zbl 0891.47040
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