my forty years on his shoulders - Department of Mathematics

44A selection for S is a selection for S on R.We say that S is symmetric if and only if S(x,y) ↔ S(y,x).THEOREM 9.1. Let S ⊆ R 2 be a symmetric Borel set. Then S or R 2 \S has a Borelselection.My proof of Theorem 9.1 in (Friedman 1981) relied heavily on Boreldeterminacy, due to D.A. Martin. See (Martin 1975), (Martin 1985), and (Kechris1994 137-148).THEOREM 9.2. (Friedman 1981). Theorem 9.1 is provable in ZFC, but notwithout the axiom scheme of replacement.There is another kind of Borel selection theorem that is implicit in work ofDebs and Saint Raymond of Paris VII. They take the general form: if there is anice selection for S on compact subsets of E, then there is a nice selection for S onE. See the five papers of Debs and Saint Raymond in the references.THEOREM 9.3. Let S ⊆ R 2 be Borel and E ⊆ R be Borel with empty interior. Ifthere is a continuous selection for S on every compact subset of E, then there is acontinuous selection for S on E.THEOREM 9.4. Let S ⊆ R 2 be Borel and E ⊆ R be Borel. If there is a Borelselection for S on every compact subset of E, then there is a Borel selection for Son E.