Partially ordered space
In mathematics, a partially ordered space[1] (or pospace) is a topological space [math]\displaystyle{ X }[/math] equipped with a closed partial order [math]\displaystyle{ \leq }[/math], i.e. a partial order whose graph [math]\displaystyle{ \{(x, y) \in X^2 \mid x \leq y\} }[/math] is a closed subset of [math]\displaystyle{ X^2 }[/math].
From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.
Equivalences
For a topological space [math]\displaystyle{ X }[/math] equipped with a partial order [math]\displaystyle{ \leq }[/math], the following are equivalent:
- [math]\displaystyle{ X }[/math] is a partially ordered space.
- For all [math]\displaystyle{ x,y\in X }[/math] with [math]\displaystyle{ x \not\leq y }[/math], there are open sets [math]\displaystyle{ U,V\subset X }[/math] with [math]\displaystyle{ x\in U, y\in V }[/math] and [math]\displaystyle{ u \not\leq v }[/math] for all [math]\displaystyle{ u\in U, v\in V }[/math].
- For all [math]\displaystyle{ x,y\in X }[/math] with [math]\displaystyle{ x \not\leq y }[/math], there are disjoint neighbourhoods [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x }[/math] and [math]\displaystyle{ V }[/math] of [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ U }[/math] is an upper set and [math]\displaystyle{ V }[/math] 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 [math]\displaystyle{ = }[/math] as the partial order, this definition becomes the definition of a Hausdorff space.
Since the graph is closed, if [math]\displaystyle{ \left( x_{\alpha} \right)_{\alpha \in A} }[/math] and [math]\displaystyle{ \left( y_{\alpha} \right)_{\alpha \in A} }[/math] are nets converging to x and y, respectively, such that [math]\displaystyle{ x_{\alpha} \leq y_{\alpha} }[/math] for all [math]\displaystyle{ \alpha }[/math], then [math]\displaystyle{ x \leq y }[/math].
See also
- Ordered vector space – Vector space with a partial order
- Ordered topological vector space
- Topological vector lattice
References
- ↑ Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S. (2009). Continuous Lattices and Domains. doi:10.1017/CBO9780511542725. ISBN 9780521803380.
- 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
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