Ordinal Decompositions and Extreme Selections
Valentin Gutev
TL;DR
The paper develops two natural constructions of extreme hyperspace selections via ordinal decompositions, employing sets clopen modulo a point to streamline arguments and derive new results. It introduces ordinal and quasi-ordinal decompositions to generate point-maximal and point-minimal selections for $\mathscr{F}(X)$, with continuity guaranteed under careful limit-ordinal conditions. A selection relation $\trianglelefteq_f$ and the associated clopen-modulo-a-point structures underpin simple proofs and gluing techniques, while cut points yield zero-dimensionality and first countability at critical points. The work extends these ideas to general decompositions, provides direct proofs for classical results, and applies them to the behavior of $\beta X$ in the realm of orderability and to pseudocompact spaces, concluding with natural questions about the limits of these techniques.
Abstract
The paper contains two natural constructions of extreme hyperspace selections generated by special ordinal decompositions of the underlying space. These constructions are very efficient not only in simplifying arguments but also in clarifying the ideas behind several known results. They are also crucial in obtaining some new results for such extreme selections. This is achieved by using special sets called clopen modulo a point. Such sets are naturally generated by a relation between closed sets and points of the space with respect to a given hyperspace selection.