Topological properties of Souslin subsets

Let $X$ be a subparacompact regular space such that the projection map $p:X \times {\mathbf{P}} \to X$, where ${\mathbf{P}}$ is the space of irrational numbers, preserves collections of sets having a $\sigma$-locally finite refinement. It is shown that $p$ then preserves generalized ${F_\sigma }$-sets. It follows that, if $X$ has any tpological property which is hereditary with respect to generalized ${F_\sigma }$-sets, then every Souslin subset of $X$ will also have this property in the relative topology. Such topological properties include nearly all covering properties (paracompactness, metacompactness, etc.), as well as normality, collectionwise normality, and the Lindelöf property. We show that the above mapping property will hold whenever $X$ is a $P(\omega )$-space, thus, in particular, when $X$ is any Souslin (hence any Baire) subset of a compact space crossed with a metrizable space. Additional topological properties of Souslin subsets, such as topological completeness, realcompactness, $z$-embeddedness, and the properties of being a $P$-space or $\Sigma$-space, are also considered.