Near coherence of filters. II. Applications to operator ideals, the Stone-Čech remainder of a half-line, order ideals of sequences, and slenderness of groups

Abstract

The set-theoretic principle of near coherence of filters (NCF) is known to be neither provable nor refutable from the usual axioms of set theory. We show thatNCF is equivalent to the following statements, among others: (1) The ideal ofcompact operators on Hilbert space is not the sum of two smaller ideals. (2) TheStone-Cech remainder of a half-line has only one composant. (This was first provedby J. Mioduszewski.) (3) The partial ordering of slenderness classes of abeliangroups, minus its top element, is directed upward (and in fact has a top element).Thus, all these statements are also consistent and independent. © 1987 American Mathematical Society.