nLab
completely prime filter

Recall that a filter $F$ on a lattice $L$ is called prime if $\perp \notin F$ and, whenever $x\vee y\in F$, then $x\in F$ or $y\in F$. In other words, for every finite index set $I$, $x_k\in F$ for some $k$ whenever ${\bigvee }_{k:I}{x}_i\in F$.

We now generalise from finitary joins to arbitrary joins: A filter $F$ on a complete lattice $L$ is completely prime if, for any index set $I$ whatsoever, $x_k\in F$ for some $k$ whenever ${\bigvee }_{k:I}{x}_i\in F$. Equivalently, a completely prime filter is given by a simlutaneous suplattice and lattice homomorphism from $L$ to the lattice $\mathrm{TV}$ of truth values (which is classically the boolean domain $𝟚$).

In particular, if $L$ is a frame, then a completely prime filter of $L$ is given by a frame homomorphism from $L$ to $\mathrm{TV}$. Thinking of $L$ as a locale, this is the same as a point of $L$.