Partially ordered space

In mathematics, a partially ordered space (or pospace) is a topological space $$X$$ equipped with a closed partial order $$\leq$$, i.e. a partial order whose graph $$\{(x, y) \in X^2 \mid x \leq y\}$$ is a closed subset of $$X^2$$.

From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.

Equivalences
For a topological space $$X$$ equipped with a partial order $$\leq$$, the following are equivalent: The order topology is a special case of this definition, since a total order is also a partial order.
 * $$X$$ is a partially ordered space.
 * For all $$x,y\in X$$ with $$x \not\leq y$$, there are open sets $$U,V\subset X$$ with $$x\in U, y\in V$$ and $$u \not\leq v$$ for all $$u\in U, v\in V$$.
 * For all $$x,y\in X$$ with $$x \not\leq y$$, there are disjoint neighbourhoods $$U$$ of $$x$$ and $$V$$ of $$y$$ such that $$U$$ is an upper set and $$V$$ is a lower set.

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 $$\left( x_{\alpha} \right)_{\alpha \in A}$$ and $$\left( y_{\alpha} \right)_{\alpha \in A}$$ are nets converging to x and y, respectively, such that $$x_{\alpha} \leq y_{\alpha}$$ for all $$\alpha$$, then $$x \leq y$$.