Partially ordered space

In mathematics, a partially ordered space^[1] (or pospace) is a topological space ${\displaystyle X}$ equipped with a closed partial order ${\displaystyle \leq }$, i.e. a partial order whose graph ${\displaystyle \{(x,y)\in X^{2}\mid x\leq y\}}$ is a closed subset of ${\displaystyle X^{2}}$.

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

Equivalences

For a topological space ${\displaystyle X}$ equipped with a partial order ${\displaystyle \leq }$, the following are equivalent:

• ${\displaystyle X}$ is a partially ordered space.
• For all ${\displaystyle x,y\in X}$ with ${\displaystyle x\not \leq y}$, there are open sets ${\displaystyle U,V\subset X}$ with ${\displaystyle x\in U,y\in V}$ and ${\displaystyle u\not \leq v}$ for all ${\displaystyle u\in U,v\in V}$.
• For all ${\displaystyle x,y\in X}$ with ${\displaystyle x\not \leq y}$, there are disjoint neighbourhoods ${\displaystyle U}$ of ${\displaystyle x}$ and ${\displaystyle V}$ of ${\displaystyle y}$ such that ${\displaystyle U}$ is an upper set and ${\displaystyle V}$ 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 ${\displaystyle =}$ as the partial order, this definition becomes the definition of a Hausdorff space.

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

See also

• Ordered vector space – Vector space with a partial order
• Ordered topological vector space
• Topological vector lattice