Partially ordered space

In mathematics, a partially ordered space{{cite book|title=Continuous Lattices and Domains|last1=Gierz|first1=G.|last2=Hofmann|first2=K. H.|last3=Keimel|first3=K.|last4=Lawson|first4=J. D.|last5=Mislove|first5=M.|last6=Scott|first6=D. S.|year=2009|doi=10.1017/CBO9780511542725|isbn=9780521803380 }} (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 Beckenstein Topological Vector Spaces|edition=2}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}

External links

[http://planetmath.org/orderedspace ordered space] on Planetmath

Category:Topological spaces

Partially ordered space

Equivalences

Properties

See also

References

External links