# Partially ordered space

## From Wikipedia, the free encyclopedia

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.

## 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 .

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