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.

be two elements of a preordered set

, if the following two equivalent conditions are satisfied.

, 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.

is an upper set, and

a lower set.

is a directed set (that is,

A preordered set

is called a continuous preordered set if for any

of a continuous preordered set

if and only if for any directed set

From this follows the interpolation property of the continuous preordered set

of a continuous dcpo

, the following two conditions are equivalent.

[1]: p.61, Proposition I-1.19(i) Using this it can be shown that the following stronger interpolation property is true for continuous dcpos.

[1]: p.61, Proposition I-1.19(ii) For a dcpo

, the following conditions are equivalent.

[1]: Theorem I-1.10 In this case, the actual left adjoint is For any two elements

of a complete lattice

, there is a finite subset

be a complete lattice.

Then the following conditions are equivalent.

A continuous complete lattice is often called a continuous lattice.

For a topological space

, the following conditions are equivalent.