Partially ordered space


In mathematics, a partially ordered space is a topological space equipped with a closed partial order, i.e. a partial order whose graph is a closed subset of.
From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.

Equivalences

For a topological space equipped with a partial order, 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 as the partial order, this definition becomes the definition of a Hausdorff space.
Since the graph is closed, if and are nets converging to x and y, respectively, such that for all, then.