An injective characterization of Peano spaces

It is shown that a continuous map defined on a closed zero-dimensional subspace S of a compact space T into a Peano space X can be continuously extended over T or, equivalently, X is an AE(0, ∞),and this property precisely characterizes Peano spaces within the class of compact metric spaces. Surjectively, a compact AE(0, ∞) of arbitrary weight is proved to be the continuous image of a Tychonoff cube by a map satisfying the zero-dimensional lifting property.