Abstract
We survey current stage of effective descriptive set theory which was fast evolving in the last decade. Most attention will be given to (effective) quasi-Polish spaces and their important subclasses (Polish spaces, ω-continuous domains) but also some important non-countably-based spaces (like the Kleene-Kreisel functionals or spaces of polynomials) will be discussed. Along with effective DST, also classical DST will be considered, which is the “limit” of the effective DST relative to arbitrary oracles. We will discuss not only classical hierarchies of sets and functions but also versions of (effective) Wadge hierarchy where a considerable progress was recently achieved.
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