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 lnown to be neither provable nor refutable from the usual axioms of set theory. We show that NCF is equivalent to the following statements, among others: (1) The ideal of compact operators on Hilbert space is not the sum of two smaller ideals. (2) The Stone-tech remainder of a half-line has only one composant. (This was first proved by J. Mioduszewski.) (3) The partial ordering of slenderness classes of abelian groups, minus its top element, is directed upward (and in fact has a top element). Thus, 11 these statements are also consistent and independent.