Nonuniformization results for the projective hierarchy

Let X and Y be uncountable Polish spaces. We show in ZF that there is a coanalytic subset P of X x Y with countable sections which cannot be expressed as the union of countably many partial coanalytic, or even PCA = I 2, graphs. If X = Y = co>, P may be taken to be 17 . Assuming stronger set theoretic axioms, we identify the least pointclass such that any such coanalytic P can be expressed as the union of countably many graphs in this pointclass. This last result is extended (under suitable hypotheses) to all levels of the projective hierarchy. Introduction. Let X and Y be uncountable Polish spaces. It is a well-known result of Novikov and Kondo that any Il (i.e., coanalytic) subset P of X x Y can be uniformized by a HI relation P' c P. That is, Vx E X []y E Y P(x, y) +-+ 3 a unique y P'(x, y)]. Although uniformization fails for 2I (analytic) sets, Lusin (see [Lu, p. 247]) and Novikov [No] did obtain the result that every analytic (Borel) set P c X x Y with countable sections can be written as a countable union of analytic (Borel) graphs, i.e., P = U ) GQ where Gn E 2I (A ') is a graph (throughout this paper, "graph" will denote a partial graph, i.e., Vx 3 at most one y Gn(x, y)). Similarly, assuming Aln determinacy, each P E 2+ 1 can be written as a countable union of 2:+ graphs. On the other hand, if P is HI, then P can be expressed as the union of w,), Borel sets Ba, a < w1. If, in addition, each section of P is countable, then each set Ba has countable sections and can therefore be expressed as the union of countably many Al graphs. Thus, each HI set P with countable sections can be expressed as the union of w, Borel graphs. A natural question then, raised by Mauldin, is the following. Question. Can every HI set P c X x Y with countable sections be written as the countable union of HI graphs, P = tJn Gn? We show, by working in ZF, that the answer is no in a strong way-our Theorem 1. We are grateful to W. H. Woodin for pointing out to us that our original result (which ruled out coverings by I1 graphs) could be extended to include graphs Received May 7, 1990; revised August 15, 1990. 1Research supported by National Science Foundation Grant DMS-90-07808. 2Research supported by National Science Foundation Grant DMS-88-03361. 1980 Mathematics Subject Classification (1985 Revision). Primary 04A15, 03E60; Secondary 28A05.