On the Borel class of the derived set operator

Abstract

— KURATOWSKI showed that the derived set operator D, acting on the space of closed subsets of the Cantor space 2 , is a Borel map of class exactly two and posed the problem of determining the precise classes of the higher order derivatives In part I of our work [Bull. Soc. Math. France, 110, 4, 1982, p. 357-380], we obtained upper and lower bounds for the Borel class of Z> and in particular showed that for limit ordinals , is exactly of class X+l. The first author recently showed, using different methods {cf. [1]) that for finite n, is exactly of Borel class 2 n. We now complete the solution of KURATOWSKTS problem by showing that for any limit ordinal and any finite n, the operator D^ is of Borel class exactly ^.+2n+1. In this paper, we determine the exact Borel classes of the iterated derived set operators £>, acting on the space Jf of closed subsets of the Cantor space 2^ with the usual Vietoris topology. This completes the solution of the problem of KURATOWSKI [3] which was begun in part I of our work [2]. (*) Textc rccu Ie 13 deccmbre 1982, revise Ie 10 novembrc 1983. Douglas CENZER, University of Florida; R. Daniel MAULDIN, North Texas state University, Department of Mathematics, P.O. Box 5117. Denton, Texas 76203, U.S.A. This author gratefully acknowledges support for this work under NSF grant MCS 83-01581 and a research grant from North Texas State University. BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE — 0037-9484/1983/04 367 06/S 2.60 © Gauthier-Villars 368 D. CENZER AND R. D. MAULDIN The results in the present paper depend strongly on those of its predecessor. We begin with some basic definitions and results from [2]. The derived set operator D maps ^f into Jf and is defined by : D(F)=F^{x:xeC\(F^{x})}. The a'th iterate D of the derived set operator map be defined for all ordinals a by letting D°(F)=F, D^^F) =D(D(F)) for alia and D^(F) = }{D(F) : OL w) , x(n)=0}. If y is identified with the family ^ ( N ) of subsets ofN, then S corresponds to the family of finite sets. Let 0==(0,0,0, . . .). The stitching operator mapping v into Jf is defined as follows: