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.

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 ${\displaystyle x\leq y}$ .

• 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.

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