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 $\mathbf{\Phi }(\mathcal{Z},X)$ generated from zero sets in $X$ by $\mathbf{\Phi }$ has the reduction or separation property if the corresponding class $\mathbf{\Phi }(\mathcal{F},\mathbb{R})$ 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 ${\mathbf{\Sigma }}_{2n}^1(\mathcal{Z},X)$ and ${\mathbf{\Pi }}_{2n+1}^1(\mathcal{Z},X)$ have the reduction property while ${\mathbf{\Pi }}_{2n}^1(\mathcal{Z},X)$ and ${\mathbf{\Sigma }}_{2n+1}^1(\mathcal{Z},X)$ have the separation property.