In mathematics, a partially ordered space (or pospace) 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: is a partially ordered space. For all with , there are open sets with and for all . For all with , there are disjoint neighbourhoods of and of such that is an upper set and is a lower set. 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 . See also References External links ordered space on Planetmath Topological spaces