Directed preorders and nets

A directed preorder is a nonempty set $D$ with a reflexive, transitive relation $\le$ such that every $d,e\in D$ have a common upper bound: some $f\in D$ satisfies $d\le f$ and $e\le f$. Antisymmetry is not required; thus this is a preorder obtained by omitting antisymmetry from the partial-order axioms of Partial order and partially ordered set.

If $X$ is the underlying set of a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), a net in $X$ indexed by $D$ is a function $x:D\to X$, written $(x_d{)}_{d\in D}$. The order on $D$ records which indices are sufficiently far along; it need not be a linear order.

Some texts require a directed set to be a partial order. The present preorder convention is deliberate: none of the convergence arguments needs antisymmetry, and it permits convenient index systems with equivalent stages.