On reduction and separation of projective sets in Tychonoff spaces

Abstract

We show that for every Tychonoff space $X$ and Hausdorff operation $\mathbf\Phi$, the class $\mathbf\Phi(\mathscr Z,X)$ generated from zero sets in $X$ by $\mathbf\Phi$ has the reduction or separation property if the corresponding class $\mathbf\Phi(\mathscr F,\mathbb R)$ of sets of reals has the same property. In particular, under Projective Determinacy, these properties of such projective sets in $X$ have the same pattern as the First Periodicity Theorem states for projective sets of reals: the classes $\mathbf\Sigma^{1}_{2n}(\mathscr Z,X)$ and $\mathbf\Pi^{1}_{2n+1}(\mathscr Z,X)$ have the reduction property while $\mathbf\Pi^{1}_{2n}(\mathscr Z,X)$ and $\mathbf\Sigma^{1}_{2n+1}(\mathscr Z,X)$ have the separation property.