  All spaces are assumed to be separable and metrizable. We give a complete
classification of the homogeneous zero-dimensional spaces, under the Axiom of
Determinacy. This classification is expressed in terms of topological
complexity (in the sense of Wadge theory) and Baire category. In the same
spirit, we also give a complete topological classification of the filters on
$\omega$. As by-products, we obtain purely topological characterizations of the
semifilters and filters on $\omega$. The Borel versions of these results are in
almost all cases due to Fons van Engelen. Along the way, we obtain
Wadge-theoretic results of independent interest, especially regarding closure
properties.
