A few remarks on the set of finite-to-one maps of the Cantor set. (English) Zbl 0801.54013

In this interesting paper the author discusses \(C(2^ \infty, I^ \infty)\), the space of all continuous functions from the Cantor set \(2^ \infty\) to the Hilbert cube \(I^ \infty\), endowed with the topology of uniform convergence. The subset of \(C(2^ \infty, I^ \infty)\) consisting of all finite-to-one functions is denoted by \({\mathcal C}\). The author introduces a natural Lusin-Sierpiński index for \({\mathcal C}\) and proves (among other things) that if \(f\in {\mathcal C}\) and \(X= f[2^ \infty]\) then \(\text{ind } X\leq \text{ord }f\). He also shows that there is a function \(\Psi: \omega_ 1\to \omega_ 1\) such that for each compactum \(X\subseteq I^ \infty\) without isolated points we have: If \(\text{ind } X\leq \alpha\) then there exists an element \(f\in {\mathcal C}\) with \(\text{ord } f\leq \Psi(\alpha)\) and \(f [2^ \infty] =X\).


54C35 Function spaces in general topology
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