Continuous poset

Let a,b\in P be two elements of a preordered set (P,\lesssim). Then we say that a approximates b, or that a is way-below b, if the following two equivalent conditions are satisfied. • For any directed set D\subseteq P such that b\lesssim\sup D, there is a d\in D such that a\lesssim d. • For any ideal I\subseteq P such that b\lesssim\sup I, a\in I. If a approximates b, we write a\ll b. The approximation relation \ll is a transitive relation that is weaker than the original order, also antisymmetric if P is a partially ordered set, but not necessarily a preorder. It is a preorder if and only if (P,\lesssim) satisfies the ascending chain condition. For any a\in P, let :\mathop\Uparrow a=\{b\in L\mid a\ll b\} :\mathop\Downarrow a=\{b\in L\mid b\ll a\} Then \mathop\Uparrow a is an upper set, and \mathop\Downarrow a a lower set. If P is an upper-semilattice, \mathop\Downarrow a is a directed set (that is, b,c\ll a implies b\vee c\ll a), and therefore an ideal. A preordered set (P,\lesssim) is called a continuous preordered set if for any a\in P, the subset \mathop\Downarrow a is directed and a=\sup\mathop\Downarrow a. == Properties ==

The interpolation property For any two elements a,b\in P of a continuous preordered set (P,\lesssim), a\ll b if and only if for any directed set D\subseteq P such that b\lesssim\sup D, there is a d\in D such that a\ll d. From this follows the interpolation property of the continuous preordered set (P,\lesssim): for any a,b\in P such that a\ll b there is a c\in P such that a\ll c\ll b. Continuous dcpos For any two elements a,b\in P of a continuous dcpo (P,\le), the following two conditions are equivalent. A continuous complete lattice is often called a continuous lattice. == Examples ==

Lattices of open sets For a topological space X, the following conditions are equivalent. • The complete Heyting algebra \operatorname{Open}(X) of open sets of X is a continuous complete Heyting algebra. • The sobrification of X is a locally compact space (in the sense that every point has a compact local base) • X is an exponentiable object in the category \operatorname{Top} of topological spaces. That is, the functor (-)\times X\colon\operatorname{Top}\to\operatorname{Top} has a right adjoint. == References ==