In mathematics, a partially ordered space (or pospace) is a topological space X {\displaystyle X} equipped with a closed partial order ≤ {\displaystyle \leq } , i.e. a partial order whose graph { ( x , y ) ∈ X 2 ∣ x ≤ y } {\displaystyle \{(x,y)\in X^{2}\mid x\leq y\}} is a closed subset of X 2 {\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 X {\displaystyle X} equipped with a partial order ≤ {\displaystyle \leq } , the following are equivalent:
X {\displaystyle X} is a partially ordered space. For all x , y ∈ X {\displaystyle x,y\in X} with x ≰ y {\displaystyle x\not \leq y} , there are open sets U , V ⊂ X {\displaystyle U,V\subset X} with x ∈ U , y ∈ V {\displaystyle x\in U,y\in V} and u ≰ v {\displaystyle u\not \leq v} for all u ∈ U , v ∈ V {\displaystyle u\in U,v\in V} . For all x , y ∈ X {\displaystyle x,y\in X} with x ≰ y {\displaystyle x\not \leq y} , there are disjoint neighbourhoods U {\displaystyle U} of x {\displaystyle x} and V {\displaystyle V} of y {\displaystyle y} such that U {\displaystyle U} is an upper set and V {\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 ( x α ) α ∈ A {\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}} and ( y α ) α ∈ A {\displaystyle \left(y_{\alpha }\right)_{\alpha \in A}} are nets converging to x and y, respectively, such that x α ≤ y α {\displaystyle x_{\alpha }\leq y_{\alpha }} for all α {\displaystyle \alpha } , then x ≤ y {\displaystyle x\leq y} .
See also Ordered vector space – Vector space with a partial order Ordered topological vector space Topological vector lattice
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.
External links ordered space on Planetmath