Partially ordered space

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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 | 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[edit]

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. Every pospace is a Hausdorff space. If we take equality = as the partial order, this definition becomes the definition of a Hausdorff space.

See also[edit]