Continuous poset
In order theory, a continuous poset is a partially ordered set in which every element is the directed supremum of elements approximating it. Definitions 
Let  be two elements of a preordered set . Then we say that  approximates , or that  is way-below , if the following two equivalent conditions are satisfied. For any directed set  such that , there is a  such that . For any ideal  such that , . If  approximates , we write . The approximation relation  is a transitive relation that is weaker than the original order, also antisymmetric if  is a partially ordered set, but not necessarily a preorder. It is a preorder if and only if  satisfies the ascending chain condition. For any , let

Then  is an upper set, and  a lower set. If  is an upper-semilattice,  is a directed set (that is,  implies ), and therefore an ideal. A preordered set  is called a continuous preordered set if for any , the subset  is directed and . Properties

The interpolation property 
For any two elements  of a continuous preordered set ,  if and only if for any directed set  such that , there is a  such that . From this follows the interpolation property of the continuous preordered set : for any  such that  there is a  such that . Continuous dcpos 
For any two elements  of a continuous dcpo , the following two conditions are equivalent. and . For any directed set  such that , there is a  such that  and . Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any  such that  and , there is a  such that  and . For a dcpo , the following conditions are equivalent. is continuous. The supremum map  from the partially ordered set of ideals of  to  has a left adjoint. In this case, the actual left adjoint is

Continuous complete lattices 
For any two elements  of a complete lattice ,  if and only if for any subset  such that , there is a finite subset  such that . Let  be a complete lattice. Then the following conditions are equivalent. is continuous. The supremum map  from the complete lattice of ideals of  to  preserves arbitrary infima. For any family  of directed sets of , . is isomorphic to the image of a Scott-continuous idempotent map  on the direct power of arbitrarily many two-point lattices . A continuous complete lattice is often called a continuous lattice. Examples

Lattices of open sets 
For a topological space , the following conditions are equivalent. The complete Heyting algebra  of open sets of  is a continuous complete Heyting algebra. The sobrification of  is a locally compact space (in the sense that every point has a compact local base)
  is an exponentiable object in the category  of topological spaces. That is, the functor  has a right adjoint. References

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Order theory