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 ${\displaystyle \{(x,y)\in X^{2}|x\leq y\}}$ 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

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 ${\displaystyle x\not \leq y}$, 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 $x,y\in X$ with ${\displaystyle x\not \leq y}$, 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

• Ordered vector space

External links

• ordered space on Planetmath