# Partially ordered space

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

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

## Equivalences

For a topological space ${\displaystyle X}$ 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.

## 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 ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A))$ 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 ${\displaystyle x\leq y}$.

## References

1. ^ 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.
Partially ordered space

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