Covering dimension and finite-to-one maps

TL;DR

This paper explores whether Hurewicz's dimension characterization via finite-to-one maps extends to compact F-spaces of weight c, showing under CH that such spaces are images of zero-dimensional spaces via at most 2n-to-1 maps.

Contribution

It demonstrates, assuming the Continuum Hypothesis, that n-dimensional compact F-spaces of weight c are continuous images of zero-dimensional spaces with controlled finite-to-one maps.

Findings

Under CH, n-dimensional compact F-spaces are images of zero-dimensional spaces.

Such images are achieved via at most 2n-to-1 maps.

The result extends Hurewicz's dimension characterization to a broader class.

Abstract

Hurewicz' characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most 2n-to-1 map.