Continuous poset
# 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.

* 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 $condition.^{:p.52,ExamplesI-1.3,(4)}$

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 $(P,\leq )$, the following two conditions are $equivalent.^{:p.61,PropositionI-1.19(i)}$

* $a\ll b$ and $a\neq b$.
* For any directed set $D\subseteq P$ such that $b\leq \sup D$, there is a $d\in D$ such that $a\ll d$ and $a\neq d$.

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 $a\neq b$, there is a $c\in P$ such that $a\ll c\ll b$ and $a\neq c$$.^{:p.61,PropositionI-1.19(ii)}$

For a dcpo $(P,\leq )$, the following conditions are $equivalent.^{:TheoremI-1.10}$

* $P$ 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.

In this case, the actual left adjoint is

    ${\Downarrow }\colon P\to \operatorname {Ideal} (P)$
    ${\mathord {\Downarrow }}\dashv \sup$

### 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 $b\leq \sup A$, there is a finite subset $F\subseteq A$ such that $a\leq \sup F$.

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\}$$.^{:p.56,Theorem44}$

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 $spaces.^{:p.196,TheoremII-4.12}$ That is, the functor $(-)\times X\colon \operatorname {Top} \to \operatorname {Top}$ has a right adjoint.
