DESCRIPTIVE BAIRE AND DESCRIPTIVE ^-ANALYTIC SETS By ERIC JOHN BRAUDE [Received 24 July 1970—Revised 10 December 1970] 1. Introduction The introduction of the notions of a descriptive Borel set and a generalized analytic set (see for example [3] and [13]), leads to the extension to more general spaces of some aspects of the classical theory of Borel and analytic sets in complete separable metric spaces. For example, the family of analytic subsets is closed under the operation (s#); and there is a separation property, applicable to this family in completely regular spaces, which generalizes Lusin's first separation theorem. Other aspects of the classical theory have recently been extended by considering the Baire and i2f-Souslin sets. (See [9], where, for example, a version of Lusin's second separation theorem is proved.) In [11], Knowles and Rogers introduced a type of Baire set called a descriptive Baire set. In this paper, we introduce the related notion of a descriptive i2f-analytic set which is a type of i2f-Souslin set. The results that we obtain, when the underlying space is Hausdorff, parallel those for descriptive Borel and analytic sets. The author is indebted to J. E. Jayne for making available to him unpublished notes on Baire and ^-Souslin sets ([10]), and for suggesting many improvements in the preparation of this manuscript. In addition, the author wishes to express his appreciation to E. R. Lorch for his encouragement and guidance. DEFINITIONS and NOTATION. Throughout this paper, all topological spaces will be HausdorfF. Let N be the set of (strictly) positive integers with the discrete topology. Let I = N** with the product topology. For each element i = (i 1} i 2> ...) of I, and each n in N, i\n denotes the finite sequence (i v i 2 , ...,i n ). Let Q be a topological space. Given a collection {Z)(i | n): i e I, n e N], of subsets of Q, the set U i6 1 fln=i ^(* I n ) * s s a ^ to ^ e * ne resu l* of applying the operation (J^) to that collection. If Jf is a family of subsets of Q, then the Jf-Souslin (or Souslin-Jf) sets are the members of the smallest family of subsets which contains Jf and which is closed under the operation {jtf). h3^ is defined as the Proc.London Math. Soc. (3) 23 (1971) 409-27