In mathematics, a partially ordered space[1] (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.


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.


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


  1. ^ Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S. (2009). Continuous Lattices and Domains. doi:10.1017/CBO9780511542725. ISBN 9780521803380.