By Y. N. Moschovakis

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**Extra info for Descriptive Set Theory**

**Example text**

Hint. Construct homeomorphisms of o x X and X use induction on k. 3. Prove that if I = XI X X X, is a product space with at least one factor Xi not w, then I is a perfect Polish space. 4. Prove that a pointset P is 2; if and only if P= uT=oFi, with each Fi closed. Similarly, P is I:, if and only if P= with each Giopen. 71 THE BOREL POINTCLASSES OF FINITE ORDER 23 This is the classical definition of F, and G8 sets. These occur quite often in analysis, for example consider the following problem. 5.

Writing R(x, n ) w (Vm 5 n ) ( 3 s )i P(x, m, s) rather than R =V‘T 1P. In fact the whole proof rested on some quantifier manipulation rules whose truth is transparent in logical notation. We list them here for reference, but we will apply them in the future without much ado. (3s)@t)P(s,t ) W ( 3 U ) p ( ( U ) o , ( U ) I ) , (VS)(WP(S,I ) * ( V u P ( b ) o , ( U ) l ) , (Vm 5 n)(3s)P(m, s) w (3u)(Vm 5 n)P(m, (u),), (3m5 n)(Vs)P(m,s) w (Vu )(3 mIn)P(m,(u),). These rules are useful because they allow us to simplify the quantifier prefix of a complicated logical expression by introducing continuous substitutions in the matrix.

THEOREM. Let r be an X-parametrized pointclass which is closed under continuous substitution. If r is closed under 3", then it is closed under V" and if r is closed under V", then it is also closed under A". Suppose Pi G X, Pi E r, let G G X x X be universal and choose irrationals E~ such that Pi = G,, = {x E X:G ( q ,x)}. Now pick E so that for every i, (&)i = Ei and set x EP M (3i)G((s)i, x). Clearly P E r by closure under continuous substitution and 3"' and P = Ui Pi* -I The argument about V" is similar.