By Y. N. Moschovakis
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During the last fifteen years, the geometrical and topological tools of the speculation of manifolds have assumed a vital function within the such a lot complicated components of natural and utilized arithmetic in addition to theoretical physics. the 3 volumes of "Modern Geometry - tools and functions" comprise a concrete exposition of those tools including their major functions in arithmetic and physics.
One of many goals of this paintings is to enquire a few average homes of Borel units that are undecidable in $ZFC$. The authors' place to begin is the next hassle-free, although non-trivial outcome: give some thought to $X \subset 2omega\times2omega$, set $Y=\pi(X)$, the place $\pi$ denotes the canonical projection of $2omega\times2omega$ onto the 1st issue, and consider that $(\star)$ : ""Any compact subset of $Y$ is the projection of a few compact subset of $X$"".
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Prove that if f : R 'X is continuous and F is a closed set of reals, then f [ F ] is 2 : . Hint. R is a countable union of compact sets. -I Practically every specific pointset which comes up in the usual constructions of analysis and topology is easily shown to be projective-in fact, almost always, it is 2; or n:. We only mention a couple of simple examples here, since we will meet several interesting projective pointsets later on. 8. On the space C[O, 11 of continuous real functions on the unit interval, put Q ( f )w f is differentiable on [0, 11, Rcf)H f is continuously differentiable on [0, 11, where a t the endpoints we naturally take the one-sided derivatives.
We refer to any such g as a Borel inverse of f. It will turn out that every Borel injection is a good Borel injection. This is a special case of a fairly difficult theorem which we will prove in 2E and again in Chapter 4. Here we only need show that enough good Borel injections exist. Notice that if f : X y is a good Borel injection, then - Y f m1- f(dY)) = Y with g any Borel inverse of f, so f[X] is a Borel set.
4. 3 Clearly T and T are Zz by the closure properties of this pointclass. T” is X:+l. 1and since i Q is also is Z;,, by two applications of closure of this pointclass under 3”. This kind of computation is so simple that we will not usually bother to put it down. 4 above which shows step-by-step the properties of the relevant pointclasses that we use. 3. Let f :63 4(R be a continuous function on the line. Prove that the relations Y) f’(x> = Y, m, @ Q(x) e f ’ ( x )exists are both @. Hint. Let r,, r I ,...