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 $$a,b\in P$$ 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. 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 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 $$b\lesssim\sup I$$, $$a\in I$$.

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 $$a=\sup\mathop\Downarrow a$$.

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. Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any $$a,b\in P$$ such that $$a\ll b$$ and $$a\ne b$$, there is a $$c\in P$$ such that $$a\ll c\ll b$$ and $$a\ne c$$.
 * $$a\ll b$$ and $$a\ne b$$.
 * For any directed set $$D\subseteq P$$ such that $$b\le\sup D$$, there is a $$d\in D$$ such that $$a\ll d$$ and $$a\ne d$$.

For a dcpo $$(P,\le)$$, the following conditions are equivalent. In this case, the actual left adjoint is
 * $$P$$ is continuous.
 * The supremum map $$\sup \colon \operatorname{Ideal}(P)\to P$$ from the partially ordered set of ideals of $$P$$ to $$P$$ has a left adjoint.
 * $${\Downarrow} \colon P\to\operatorname{Ideal}(P)$$
 * $$\mathord\Downarrow\dashv\sup$$

Continuous complete lattices
For any two elements $$a,b\in L$$ of a complete lattice $$L$$, $$a\ll b$$ if and only if for any subset $$A\subseteq L$$ such that $$b\le\sup A$$, there is a finite subset $$F\subseteq A$$ such that $$a\le\sup F$$.

Let $$L$$ be a complete lattice. Then the following conditions are equivalent.
 * $$L$$ is continuous.
 * The supremum map $$\sup \colon \operatorname{Ideal}(L)\to L$$ from the complete lattice of ideals of $$L$$ to $$L$$ preserves arbitrary infima.
 * For any family $$\mathcal D$$ of directed sets of $$L$$, $$\textstyle\inf_{D\in\mathcal D}\sup D=\sup_{f\in\prod\mathcal D}\inf_{D\in\mathcal D}f(D)$$.
 * $$L$$ is isomorphic to the image of a Scott-continuous idempotent map $$r \colon \{0,1\}^\kappa\to\{0,1\}^\kappa$$ on the direct power of arbitrarily many two-point lattices $$\{0,1\}$$.

A continuous complete lattice is often called a continuous lattice.

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.