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From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.
For a topological space equipped with a partial order , the following are equivalent:
- is a partially ordered space.
- For all with , there are open sets with and for all .
- For all with , there are disjoint neighbourhoods of and of such that is an upper set and is a lower set.
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 and are nets converging to x and y, respectively, such that for all , then .
- 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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