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

References

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. Cite journal requires |journal= (help)

• 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

External links

• ordered space on Planetmath