  The well known ideal presentations of countably based domains were recently
extended to (effective) quasi-Polish spaces. Continuing these investigations,
we explore some classes of effective quasi-Polish spaces. In particular, we
prove an effective version of the domain-characterization of quasi-Polish
spaces, describe effective extensions of quasi-Polish topologies, discover
natural numberings of classes of effective quasi-Polish spaces, estimate the
complexity of the (effective) homeomorphism relation and of some classes of
spaces w.r.t. these numberings, and investigate degree spectra of continuous
domains.
