On a Notion of Smallness for Subsets of the Baire Space

Let us call a set A C o' of functions from X into X a-bounded if there is a countable sequence of functions (a.: n E o} C w' such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of o'. We show that most of the usual definability results about the structure of countable subsets of o' have corresponding versions which hold about a-bounded subsets of o'. For example, we show that every 42,+1 a-bounded subset of o' has a Al,n+ "bound" (am: m E W} and also that for any n > 0 there are largest a-bounded fI!n+l and 'n+2 sets. We need here the axiom of projective determinacy if n > 1. In order to study the notion of a-boundedness a simple game is devised which plays here a role similar to that of the standard *-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the *and **(or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of " whose special cases include countability, being of the first category and a-boundedness and for which one can generalize all the main results of the present paper. 1. Preliminaries. IA. Let cX = {O, 1, 2.... } be the set of all natural numbers and % = c' the set of all functions from X to X or, for simplicity, reals. Letters i, j, k, 1, m, n, . . . denote elements of cX and a, 13 y, 8, ... reals. We study subsets of the product spaces DC = X1 x X2 x * x Xk, where X is X or & We call such subsets pointsets. Sometimes we think of them as relations and we write interchangeably x E A X A(x). A pointclass is a class of pointsets, usually in all product spaces. We shall be concerned primarily in this paper with the analytical pointclasses f, HII, 1,A and their corresponding projective pointclasses S1, fH1, Al. For information about them we refer the reader to [R], [Sh] and [Mol ]. If F is a pointclass and DC a product space (DC = X or = 6R will be enough for this definition), then we say that F is D-parametrized if for any product space @ there is a G E F, G S DC x 6@ such that letting Gx = {y: (x,y) E G} we have {A 5 @: A E F) = {Gx x E DC). In this case G is called D-universal Received by the editors December 10. 1975. AMS (MOS) subject classifications (1970). Primary 04A15, 02K30, 28AO5, 54HO5; Secondary 02F35, 02KO5, 02K25, 02K35, 04A30. C Amerisan Mathematical Society 1977