Chain-complete posets and directed sets with applications

Let a poset P be called chain-complete when every chain, including the empty chain, has a sup in P. Many authors have investigated properties of posets satisfying some sort of chain-completeness condition (see [,11, [-31, [6], I-71, [17], [,181, ['191, [,211, [,221), and used them in a variety of applications. In this paper we study the notion of chain-completeness and demonstrate its usefulness for various applications. Chain-complete posets behave in many respects like complete lattices; in fact, a chaincomplete lattice is a complete lattice. But in many cases it is the existence of sup's of chains, and not the existence of arbitrary sup's, that is crucial. More generally, let P be called chain s-complete when every chain of cardinality not greater than ~ has a sup. We first show that if a poset P is chain s-complete, then every directed subset of P with cardinality not exceeding ct has a sup in P. This sharpens the known result ([,8], [,181) that in any chain-complete poset, every directed set has a sup. Often a property holds for every directed set i f and only if it holds for every chain. We show that direct (inverse) limits exist in a category if and only if 'chain colimits' ('chain limits') exist. Since every chain has a well-ordered cofinal subset [11, p. 681, one need only work with well-ordered collections of objects in a category to establish or disprove the existence of direct and inverse limits. Similarly, a topological space is compact if and only if every 'chain of points' has a cluster point. A 'chain of points' is a generalization of a sequence. Chain-complete posers, like complete lattices, arise from closure operators in a fairly direct manner. Using closure operators we show how to form the chaincompletion P of any poset P. The chain-completion/~ of a poset P is a chain-complete poset with the property that any chain-continuous map from a poser P into a chain-complete poset Q extends uniquely to a chain-continuous map from the completion/~ into Q, where by a chaincontinuous map we mean one that preserves sup's of chains. If P is already chaincomplete, then/~ is naturally isomorphic to P. This completion is not the MacNeille