Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set together with a reflexive and transitive binary relation (that is, a preorder), with the additional property that every pair of elements has an upper bound. In other words, for any and in there must exist in with and A directed set's preorder is called a direction. The notion defined above is sometimes called an . A is defined analogously, meaning that every pair of elements is bounded below. Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other authors call a set directed if and only if it is directed both upward and downward. Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast ordered sets, which need not be directed). Join-semilattices (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward. In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) . In addition to the definition above, there is an equivalent definition. A directed set is a set with a preorder such that every finite subset of has an upper bound. In this definition, the existence of an upper bound of the empty subset implies that is nonempty. The set of natural numbers with the ordinary order is one of the most important examples of a directed set (and so is every totally ordered set). By definition, a is a function from a directed set and a sequence is a function from the natural numbers Every sequence canonically becomes a net by endowing with If is a real number then the set can be turned into a directed set by defining if (so "greater" elements are closer to ). We then say that the reals have been directed towards This is an example of a directed set that is partially ordered nor totally ordered.