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Partially ordered space

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In mathematics, a partially ordered space[1] (or pospace) is a topological space equipped with a closed partial order , i.e. a partial order whose graph is a closed subset of .

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

Equivalences

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.

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 and are nets converging to x and y, respectively, such that for all , then .

See also

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.
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Partially ordered space
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