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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)
##### MSC:
 54F15 Continua and generalizations 54C10 Special maps on topological spaces (open, closed, perfect, etc.)
##### Keywords:
Peano map; value of openness