Continuous mappings from Cantor spaces onto inverse limit spectra

Abstract:

Let $\mathcal {S} = \{ {X_\alpha };{f_{\alpha \beta }}:{X_\beta } \to {X_\alpha }\}$ be an inverse limit spectrum of compact Hausdorff spaces. We obtain necessary and sufficient conditions that there be a closed subspace W of a Cantor space and a family $\{ {f_\alpha }:W \to {X_\alpha }\}$ of continuous surjections such that for each pair $\alpha < \beta ,{f_{\alpha \beta }} \circ {f_\beta } = {f_\alpha }$. This result is applied to a special class of inverse spectra.