BorelSelection093001

BorelSelection093001 - 1 SELECTION FOR BOREL RELATIONS by...

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1 SELECTION FOR BOREL RELATIONS by Harvey M. Friedman* Department of Mathematics Ohio State University friedman@math.ohio-state.edu http://www.math.ohio-state.edu/~friedman/ September 30, 2001 Abstract. We present several selection theorems for Borel relations, involving only Borel sets and functions, all of which can be obtained as consequences of closely related theorems proved in [DSR 96,99,01,01X] involving coanalytic sets. The relevant proofs given there use substantial set theoretic methods, which were also shown to be necessary. We show that none of our Borel consequences can be proved without substantial set theoretic methods. The results are established for Baire space. We give equivalents of some of the main results for the reals. Introduction. Let S be a set of ordered pairs and A be a set. We say that f is a selection for S on A if and only if dom(f) = A and for all x A, (x,f(x)) S. We say that f is a selection for S if and only if f is a selection for S on {x: ( $ y)((x,y) S)}. Let N be the set of all nonnegative integers. 2 N = {0,1} N is Cantor space, where {0,1} is given the discrete topology. N N is Baire space, where N is given the discrete topology. We use for the reals with the usual topology. All results in sections 1-5 are formulated on N N . This is most convenient for the proofs. In section 6 we give equivalent formulations on of some of the main results. The following result from [DSR01X] led to this research. PROPOSITION I. Let S Õ N N x N N be coanalytic and E Õ N N be Borel. If there is a continuous selection for S on every compact subset of E, then there is a continuous selection for S on E.
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2 Proposition I is proved in [DSR01X] using set theoretic assumptions going beyond ZFC. In fact, [DSR01X] shows that Proposition I is provably equivalent to COUNT. For all f N N , N N L[f] is countable. over ZFC. [DSR01X] also gives the comparatively simple proof of Proposition I using analytic determinacy, which works for all coanalytic E. Moreoever, [DSR01X] shows that Proposition I for coanalytic E is equivalent to analytic determinacy. The proof of Proposition I from COUNT in [DSR01X] is rather complicated, and we conjecture that the proof can be considerably simplified using additional methods from modern set theory. In section 1, we present the simple [DSR01X] proof of Proposition I from analytic determinacy. It is obvious that if “coanalytic” is replaced by “Borel”, then the proof goes through unmodified with Borel determinacy instead of analytic determinacy. Therefore we have the following theorem of ZFC. THEOREM II. Let S Õ N N x N N be Borel and E Õ N N be Borel. If there is a continuous selection for S on every compact subset of E, then there is a continuous selection for S on E. In section 4, we show that Theorem II cannot be proved using only countably many iterations of the power set operation.
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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BorelSelection093001 - 1 SELECTION FOR BOREL RELATIONS by...

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