Extreme selections for hyperspaces of topological spaces

AbstractWe study properties of Hausdorff spaces X which depend on the variety of continuous selections for their Vietoris hyperspaces F(X) of closed non-empty subsets. Involving extreme selections for F(X), we characterize several classes of connected-like spaces. In the same way, we also characterize several classes of disconnected-like spaces, for instance all countable scattered metrizable spaces. Further, involving another type of selections for F(X), we study local properties of X related to orderability. In particular, we characterize some classes of orderable spaces with only one non-isolated point