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:

$X$ 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.

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 $\left(x_{\alpha }\right)_{\alpha \in A}$ 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 $x\leq y$ .

References

Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.