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Multiplicities of Peano maps: on a less known theorem by Hurewicz. (English) Zbl 0857.54033
A continuous function \(f:I= [0,1]\to \mathbb{R}^2\) is called a Peano map if \(f(I)\) has non-empty interior in \(\mathbb{R}^2\). The multiplicity of a value \(f(x)\) is the cardinality of \(f^{-1}(f(x))\). The authors state that in 1933, Hurewicz proved that if \(f\) is finite-to-one and has only two multiplicities for its values then it cannot be a Peano map. They consider the following concept. A value of \(y\) of a map \(f:X\to Y\) between topological spaces is called a value of openness if for each \(x\in f^{-1}(y)\) and neighborhood \(U\) of \(x\) in \(X\), \(y\in\text{int }f(U)\). The theorem in this paper can now be stated: the values with the highest multiplicities cannot be values of openness of a Peano map if they lie in the interior of the image.
Reviewer: L.R.Rubin (Norman)
54F15 Continua and generalizations
54C10 Special maps on topological spaces (open, closed, perfect, etc.)