Directed set

A set $A$ equipped with a directed order. A set $A$ with partial order $\leq$ is called upwards (respectively, downwards) directed if $\leq$ (respectively, the opposite order $\geq$) is a directed order. For example, the set of all open coverings $\{\gamma\}$ of a topological space is a downwards directed set, with $\gamma'\leq\gamma''$ if $\gamma'$ is a refinement of $\gamma''$; another example of a downwards directed set is a pre-filter, that is, a family $\delta$ of non-empty sets such that if $U,V\in\delta$ then there exists a $W\in\delta$ such that $W\subset U\cap V$. The main use of directed sets (and of filters, cf. Filter) is as index sets in the definition of generalized sequences (cf. Generalized sequence) of points, or nets, in topological spaces, in the study of the convergence of such sequences, etc.

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A pre-filter is also called a filterbase.

In addition to the topological application mentioned above, directed sets play an important role in category theory, lattice theory and theoretical computer science. In category theory they occur as the indexing sets of direct and inverse systems (see System (in a category)). In computer science, data structures are often modelled by partially ordered sets in which every upwards directed subset has a least upper bound (though finite subsets often do not); see [a1], for example. In lattice theory, least upper bounds of directed subsets again play a distinctive part; see Continuous lattice, for example.

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