# 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.