Descriptive sets

Abstract A theory of descriptive Baire sets is developed for an arbitrary completely regular space. It is shown that descriptive Baire sets are Baire sets and that they form a system closed under countable union, countable intersection and intersection with a Baire set. If a descriptive Borel set (Rogers 1965) is a Baire set then it is a descriptive Baire set. If every open set is a countable union of closed sets, the descriptive Baire sets coincide with the descriptive Borel sets. It follows, in particular, that in a metric space a set is descriptive Baire, if, and only if, it is absolutely Borel and Lindelöf.