Let ${\displaystyle a,b\in P}$  be two elements of a preordered set ${\displaystyle (P,\lesssim )}$ . Then we say that ${\displaystyle a}$  approximates ${\displaystyle b}$ , or that ${\displaystyle a}$  is way-below ${\displaystyle b}$ , if the following two equivalent conditions are satisfied.

• For any directed set ${\displaystyle D\subseteq P}$  such that ${\displaystyle b\lesssim \sup D}$ , there is a ${\displaystyle d\in D}$  such that ${\displaystyle a\lesssim d}$ .
• For any ideal ${\displaystyle I\subseteq P}$  such that ${\displaystyle b\lesssim \sup I}$ , ${\displaystyle a\in I}$ .

If ${\displaystyle a}$  approximates ${\displaystyle b}$ , we write ${\displaystyle a\ll b}$ . The approximation relation ${\displaystyle \ll }$  is a transitive relation that is weaker than the original order, also antisymmetric if ${\displaystyle P}$  is a partially ordered set, but not necessarily a preorder. It is a preorder if and only if ${\displaystyle (P,\lesssim )}$  satisfies the ascending chain condition.^{[1]: p.52, Examples I-1.3, (4)}

For any ${\displaystyle a\in P}$ , let

${\displaystyle \mathop {\Uparrow } a=\{b\in L\mid a\ll b\}}$ 

${\displaystyle \mathop {\Downarrow } a=\{b\in L\mid b\ll a\}}$ 

Then ${\displaystyle \mathop {\Uparrow } a}$  is an upper set, and ${\displaystyle \mathop {\Downarrow } a}$  a lower set. If ${\displaystyle P}$  is an upper-semilattice, ${\displaystyle \mathop {\Downarrow } a}$  is a directed set (that is, ${\displaystyle b,c\ll a}$  implies ${\displaystyle b\vee c\ll a}$ ), and therefore an ideal.

A preordered set ${\displaystyle (P,\lesssim )}$  is called a continuous preordered set if for any ${\displaystyle a\in P}$ , the subset ${\displaystyle \mathop {\Downarrow } a}$  is directed and ${\displaystyle a=\sup \mathop {\Downarrow } a}$ .

The interpolation property edit

For any two elements ${\displaystyle a,b\in P}$  of a continuous preordered set ${\displaystyle (P,\lesssim )}$ , ${\displaystyle a\ll b}$  if and only if for any directed set ${\displaystyle D\subseteq P}$  such that ${\displaystyle b\lesssim \sup D}$ , there is a ${\displaystyle d\in D}$  such that ${\displaystyle a\ll d}$ . From this follows the interpolation property of the continuous preordered set ${\displaystyle (P,\lesssim )}$ : for any ${\displaystyle a,b\in P}$  such that ${\displaystyle a\ll b}$  there is a ${\displaystyle c\in P}$  such that ${\displaystyle a\ll c\ll b}$ .

Continuous dcpos edit

For any two elements ${\displaystyle a,b\in P}$  of a continuous dcpo ${\displaystyle (P,\leq )}$ , the following two conditions are equivalent.^{[1]: p.61, Proposition I-1.19(i)}

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

Using this it can be shown that the following stronger interpolation property is true for continuous dcpos. For any ${\displaystyle a,b\in P}$  such that ${\displaystyle a\ll b}$  and ${\displaystyle a\neq b}$ , there is a ${\displaystyle c\in P}$  such that ${\displaystyle a\ll c\ll b}$  and ${\displaystyle a\neq c}$ .^{[1]: p.61, Proposition I-1.19(ii)}

For a dcpo ${\displaystyle (P,\leq )}$ , the following conditions are equivalent.^{[1]: Theorem I-1.10}

• ${\displaystyle P}$  is continuous.
• The supremum map ${\displaystyle \sup \colon \operatorname {Ideal} (P)\to P}$  from the partially ordered set of ideals of ${\displaystyle P}$  to ${\displaystyle P}$  has a left adjoint.

In this case, the actual left adjoint is

${\displaystyle {\Downarrow }\colon P\to \operatorname {Ideal} (P)}$ 

${\displaystyle {\mathord {\Downarrow }}\dashv \sup }$ 

Continuous complete lattices edit

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

Let ${\displaystyle L}$  be a complete lattice. Then the following conditions are equivalent.

• ${\displaystyle L}$  is continuous.
• The supremum map ${\displaystyle \sup \colon \operatorname {Ideal} (L)\to L}$  from the complete lattice of ideals of ${\displaystyle L}$  to ${\displaystyle L}$  preserves arbitrary infima.
• For any family ${\displaystyle {\mathcal {D}}}$  of directed sets of ${\displaystyle L}$ , ${\displaystyle \textstyle \inf _{D\in {\mathcal {D}}}\sup D=\sup _{f\in \prod {\mathcal {D}}}\inf _{D\in {\mathcal {D}}}f(D)}$ .
• ${\displaystyle L}$  is isomorphic to the image of a Scott-continuous idempotent map ${\displaystyle r\colon \{0,1\}^{\kappa }\to \{0,1\}^{\kappa }}$  on the direct power of arbitrarily many two-point lattices ${\displaystyle \{0,1\}}$ .^{[2]: p.56, Theorem 44}

A continuous complete lattice is often called a continuous lattice.

Lattices of open sets edit

For a topological space ${\displaystyle X}$ , the following conditions are equivalent.

• The complete Heyting algebra ${\displaystyle \operatorname {Open} (X)}$  of open sets of ${\displaystyle X}$  is a continuous complete Heyting algebra.
• The sobrification of ${\displaystyle X}$  is a locally compact space (in the sense that every point has a compact local base)
• ${\displaystyle X}$  is an exponentiable object in the category ${\displaystyle \operatorname {Top} }$  of topological spaces.^{[1]: p.196, Theorem II-4.12} That is, the functor ${\displaystyle (-)\times X\colon \operatorname {Top} \to \operatorname {Top} }$  has a right adjoint.

• "Continuous lattice", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Core-compact space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Continuous poset at the nLab
• Continuous category at the nLab
• Exponential law for spaces at the nLab
• Continuous poset at PlanetMath.