## 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

### Keywords:

Poincaré theorem; Brouwer’s fixed point theorem

Zbl 0891.47040
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