Definition:Completely Prime Filter

Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $F$ be a proper filter in $\struct{S, \preceq}$.

$F$ is a completely prime filter if and only if:

$\forall J \subseteq S: \paren{\sup J \in F \implies F \cap J \ne \O }$

Also see

• Definition:Completely Prime Ideal

• Results about completely prime filters can be found here.

Sources

• 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter $1$: Spaces and Lattices of Open Sets, $\S 1$ Sober spaces, Definition $1.3$