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:- 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.
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.