Selectors and Scattered Spaces

Selectors on the space of all non-empty closed subsets of a topological space, equipped with the Vietoris topology, are studied. Non-archimedean P-spaces with a continuous selector are shown to be precisely scattered spaces (or topologically well-orderable ones). Several results are given for spaces with a unique accumulation point p; e.g. if the filter of its neighborhoods (without p) is a sum two free ultrafilters then the existence of a continuous selector for all such spaces is equivalent to the non-existence of a measurable cardinal.