In mathematics, a partially ordered space[1] (or pospace) is a topological space X {\displaystyle X} equipped with a closed partial order ≤ ≤ --> {\displaystyle \leq } , i.e. a partial order whose graph { ( x , y ) ∈ ∈ --> X 2 ∣ ∣ --> x ≤ ≤ --> y } {\displaystyle \{(x,y)\in X^{2}\mid x\leq y\}} is a closed subset of X 2 {\displaystyle X^{2}} .
From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.
For a topological space X {\displaystyle X} equipped with a partial order ≤ ≤ --> {\displaystyle \leq } , the following are equivalent:
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 = {\displaystyle =} as the partial order, this definition becomes the definition of a Hausdorff space.
Since the graph is closed, if ( x α α --> ) α α --> ∈ ∈ --> A {\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}} and ( y α α --> ) α α --> ∈ ∈ --> A {\displaystyle \left(y_{\alpha }\right)_{\alpha \in A}} are nets converging to x and y, respectively, such that x α α --> ≤ ≤ --> y α α --> {\displaystyle x_{\alpha }\leq y_{\alpha }} for all α α --> {\displaystyle \alpha } , then x ≤ ≤ --> y {\displaystyle x\leq y} .
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