Compact localization and co-sobriety of Smyth power spaces

Abstract

We establish a compact-localization theorem for finite iterates of the Smyth power space. Successive union maps assign to every finite-level Smyth object a canonical least compact saturated carrier in the original space, and co-sobriety at each positive finite level is equivalent to co-sobriety of the corresponding iterates over all compact saturated carriers. Consequently, if every compact saturated subspace of a $T_0$-space $X$ is stably compact---in particular, if every such subspace is Hausdorff---then $\PS^n(X)$ is co-sober for every finite $n\geq 1$. This applies to every $k_1$-Hausdorff space and hence to every Hausdorff or metrizable space. For the first iterate, we also give a direct full-Vietoris argument, independent of stable compactness, which shows that each $k$-irreducible compact saturated family is generated by its union.