I am sorry for the intervention. I read the above mentioned article by Mioduszewski (it is available at the link http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-2e6943ca-7faf-46aa-ab2d-bbc5d5f1bcff ) but it does not contain a clear proof of the above statement: 'there exists a continuous two-to-one function $F:\mathbb{R}^{2}\to\mathbb{R}^{4}$ (but it is not surjective), that is, for every $c\in F(\mathbb{R}^{2})$ there are two and only two points $z_{1}$, $z_{2}$ such that $F(z_{1})=F(z_{2})=c$.' It is not obviously that this 'statement' is really true. Why $\mathbb{R}^{4}$ for $\mathbb{R}^{2}$ ?
space1217
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