Selectors and scattered spaces☆☆Work supported by the research project “Analisi Reale” of the Italian Ministero dell'Università e della Ricerca Scientifica e Tecnologica. The fourth author was also partially supported by the grant GA ČR 201/97/0216.

AbstractSelectors on F(X) 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