Topological equivalence to a projection. (English) Zbl 1340.54029

Let \(f:\;\mathbb R^2\to \mathbb R\) be a continuous function. Suppose that, for each \(p\in \mathbb R\) belonging to the image of \(f\), the level set \(f^{-1}(p)\) is a curve (in particular, it is path connected), and this family of curves is regular. The authors show that in this case \(f\) is topologically equivalent to a projection. The proof is based on the results of W. Kaplan [Duke Math. J. 7, 154–185 (1940; Zbl 0024.19001)].


54C30 Real-valued functions in general topology
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces


Zbl 0024.19001
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