Definability in function spaces
Let \(X\), \(Y\) denote Polish spaces. Let us recall that a~set \(A\subseteq X\) is wedge reducible to \(B\subseteq Y\) if there is a~continuous function \(f:X\to Y\) such that \(A=f^{-1}(B)\). Wedge reducibility is a~preordering and the classes of equivalences induced by this preordering are called wedge degrees. The authors study the wedge degree of a~function space viewed as a~subspace of another function space. They prove that under some appropriate hypothesis the space \(L^q(X,\mu)\) and the space of all simple functions as subsets of \(L^p(X,\mu)\) are \(\Sigma^0_2\)-complete. If \(X\)~is locally compact then for a~reasonable measure~\(\mu\), \(C(X)\cap L^p(X,\mu)\) is \(\Pi^0_3\)-complete. If \(X\)~is not \(\sigma\)-compact then this space is \(\Pi^1_1\)-complete and hence not Borel. The authors consider also spaces of Riemann integrable functions and of differentiable functions. For a~fixed Polish space~\(X\), they endow \(C(X)\) with the compact-open topology which makes it Polish and study various subspaces of~\(C(X)\).
scientific article; zbMATH DE number 1848835