Acyclic monotone normality

Abstract

A space X is acyclic monotonically normal if it has a monotonically normal operator M〈·,·〉 such that for distinct points x 0,…, x n −1 in X and x n = x 0, ∩ n −1 i = 0 M〈 x i , X⧹{ x i+1 }〉 = Ø. It is a property which arises from the study of monotone normality and the condition “chain (F)”. In this paper it is shown that GO, metric, stratifiable and elastic spaces are all acyclic monotonically normal. In addition it is established that this property is preserved by closed continuous maps, adjunction and domination. It is known that acyclic monotonically normal spaces are K 0-spaces, this being an open question for monotonically normal spaces. The links between acyclic monotone normality, monotone normality and K 0-spaces are further investigated. In particular it is shown that the addition of a simple condition to the definition of a K 0-space yields a property, called monotonically K 0, which is equivalent to acyclic monotone normality.