Partially ordered space

In mathematics, a partially ordered space (or pospace) is a topological space formula_1 equipped with a closed partial order formula_2, i.e. a partial order whose graph formula_3 is a closed subset of formula_4.
From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.
Equivalences.
For a topological space formula_1 equipped with a partial order formula_2, the following are equivalent:
The order topology is a special case of this definition, since a total order is also a partial order.
Properties.
Every pospace is a Hausdorff space. If we take equality formula_22 as the partial order, this definition becomes the definition of a Hausdorff space.
Since the graph is closed, if formula_23 and formula_24 are nets converging to "x" and "y", respectively, such that formula_25 for all formula_26, then formula_27.