Partially ordered space

In mathematics, a partially ordered space^[1] (or pospace) is a topological space ${\displaystyle X}${\displaystyle X} equipped with a closed partial order ${\displaystyle \leq }${\displaystyle \leq }, i.e. a partial order whose graph ${\displaystyle \{(x,y)\in X^{2}\mid x\leq y\}}${\displaystyle \{(x,y)\in X^{2}\mid x\leq y\}} is a closed subset of ${\displaystyle X^{2}}${\displaystyle X^{2}}.

From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.

For a topological space ${\displaystyle X}${\displaystyle X} equipped with a partial order ${\displaystyle \leq }${\displaystyle \leq }, the following are equivalent:

• ${\displaystyle X}${\displaystyle X} is a partially ordered space.
• For all ${\displaystyle x,y\in X}${\displaystyle x,y\in X} with ${\displaystyle x\not \leq y}${\displaystyle x\not \leq y}, there are open sets ${\displaystyle U,V\subset X}${\displaystyle U,V\subset X} with ${\displaystyle x\in U,y\in V}${\displaystyle x\in U,y\in V} and ${\displaystyle u\not \leq v}${\displaystyle u\not \leq v} for all ${\displaystyle u\in U,v\in V}${\displaystyle u\in U,v\in V}.
• For all ${\displaystyle x,y\in X}${\displaystyle x,y\in X} with ${\displaystyle x\not \leq y}${\displaystyle x\not \leq y}, there are disjoint neighbourhoods ${\displaystyle U}${\displaystyle U} of ${\displaystyle x}${\displaystyle x} and ${\displaystyle V}${\displaystyle V} of ${\displaystyle y}${\displaystyle y} such that ${\displaystyle U}${\displaystyle U} is an upper set and ${\displaystyle V}${\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.

Every pospace is a Hausdorff space. If we take equality ${\displaystyle =}${\displaystyle =} as the partial order, this definition becomes the definition of a Hausdorff space.

Since the graph is closed, if ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}} and ${\displaystyle \left(y_{\alpha }\right)_{\alpha \in A}}${\displaystyle \left(y_{\alpha }\right)_{\alpha \in A}} are nets converging to x and y, respectively, such that ${\displaystyle x_{\alpha }\leq y_{\alpha }}${\displaystyle x_{\alpha }\leq y_{\alpha }} for all ${\displaystyle \alpha }${\displaystyle \alpha }, then ${\displaystyle x\leq y}${\displaystyle x\leq y}.

• Ordered vector space – Vector space with a partial order
• Ordered topological vector space
• Topological vector lattice

• 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.

• ordered space on Planetmath

• Topology stubs
• Topological spaces

• Articles with short description
• Short description is different from Wikidata
• All stub articles