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# Partially ordered space

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

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

## Equivalences

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

• $X$ is a partially ordered space.
• For all $x,y\in X$ with $x\not \leq y$ , there are open sets $U,V\subset X$ with $x\in U,y\in V$ and $u\not \leq v$ for all $u\in U,v\in V$ .
• For all $x,y\in X$ with $x\not \leq y$ , there are disjoint neighbourhoods $U$ of $x$ and $V$ of $y$ such that $U$ is an upper set and $V$ 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 $=$ as the partial order, this definition becomes the definition of a Hausdorff space.

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

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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