Covering dimension and finite-to-one maps (Q649612)

The starting point of this paper is the theorem of \textit{W. Hurewicz} that the covering dimension of a separable metrizable space \(X\) is at most \(n< \infty\) iff \(X\) is the image of a zero-dimensional separable metrizable space under a closed at most \((n+1)\)-to-\(1\) map, see [``Über stetige Bilder von Punktmengen'', Proceedings Amsterdam 29, 1014--1017 (1926; JFM 52.0595.03)]. Let \(X\) be a compact \(F\)-space of weight \(\mathfrak{c}\). Assuming the Continuum Hypothesis, the authors prove, inter alia, that \((1)\) \(\dim X = \text{ind} X = \text{Ind} X\) and \((2)\) if \(\dim X= n< \infty\), then \(X\) is the continuous image of a zero-dimensional compact Hausdorff space under an at most \(2^n\)-to-\(1\) map. It is an open question whether the stronger version of \((2)\) with \(f\) at most \((n+1)\)-to-\(1\) holds.