Topologies Associated with Kuratowski-Painlevé Convergence of Closed Sets

The main purpose of this paper is to identify topologies on the closed subsets $\mathscr{C}(X)$ of a Hausdorff space X that are sequentially equivalent to classical Kuratowski-Painlevé convergence K. This reduces to a study of upper topologies sequentially equivalent to upper Kuratowski-Painlevé convergence K$^+$, where we are of course led to consider the sequential modification of upper Kuratowski-Painlevé convergence. We characterize those miss topologies induced by a cobase of closed sets that are sequentially equivalent to K$^+$, with special attention given to X first countable. Separately in the final section, we revisit Mrowka's theorem on the compactness of Kuratowski-Painlevé convergence.