In mathematics, a partially ordered space (or pospace) is a topological space X equipped with a closed partial order ≤ , i.e. a partial order whose graph { ( x , y ) ∈ X 2 | x ≤ y } is a closed subset of X 2 .
From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.
For a topological space X equipped with a partial order ≤ , the following are equivalent:
X is a partially ordered space.For all x , y ∈ X with x ≰ y , there are open sets U , V ⊂ X with x ∈ U , y ∈ V and u ≰ v for all u ∈ U , v ∈ V .For all x , y ∈ X with x ≰ y , there are disjoint neighbourhoods U of x and V of y such that U is an upper set and V is a lower set.The order topology is a special case of this definition, since a total order is also a partial order. Every pospace is a Hausdorff space. If we take equality = as the partial order, this definition becomes the definition of a Hausdorff space.
