On nonisomorphic analytic sets

Abstract

It is shown that if A is an analytic subset of I, the unit interval, such that I A is uncountable and does not contain a perfect set, then A is not Borel isomorphic to I x A or to A', n > 1, or to U, where U is a universal analytic subset of J2. It is also shown that U is not isomorphic to IxA or toA', n > 1. We shall say that two Borel structures (or measurable spaces) (X,Y:) and (Y,96) are isomorphic provided there is a one-to-one measurable map of X onto Y whose inverse is also measurable. It can be seen that if X is isomorphic to a subset Z of Y, where Z E C, and Z has the restricted Borel structure, and Y is isomorphic to a subset E of X, where E E , and E has the restricted Borel structure, then (X, :) and (Y, 6,) are isomorphic. It is well known that if B1 and B2 are Borel subsets of Polish spaces provided with the relative Borel structure, then B1 and B2 are isomorphic if and only if they have the same cardinality. The problem of the number of isomorphism classes of analytic, nonborelian subsets of Polish spaces seems to be unsolved. In [1], A. Maitra and C. RyllNardzewski show that (i) any two universal analytic sets are isomorphic, and (ii) if A is an analytic set whose complement is uncountable and does not contain a perfect set, then A is not in the isomorphic class of the universal analytic sets. In this note we give some corollaries of the techniques employed in [1], and recount their main argument in Theorem 1. First, let us set some notation. The unit interval will be denoted by I and A will denote an analytic subset of I whose complement is uncountable and does not contain a perfect set. The existence of such a set is implied by Godel's Axiom of Constructibility [2]. The n-fold product of A with itself is denoted by A'. The symbol U will denote a universal analytic subset of I x I. The dyadic rationals are denoted by Rol The binary sieve of Lebesgue is used throughout this paper [3, p. 34]. In this note, we show that no two of the following sets are isomorphic: A, I x A, and U. We show that U and A', n > 1, are not isomorphic. Also, A Received by the editors August 26, 1975 and, in revised form, November 10, 1975. AMS (MOS) subject classifications (1970). Primary 02K30, 54H05; Secondary 28A05.