Definition:Point of Locale

Definition

Let $\struct{L, \preceq}$ be a locale.

Point As Completely Prime Filter

A point of $L$ is a completely prime filter $F \subseteq L$

Point As Frame Homomorphism

A point of $L$ is a frame homomorphism $h: L \to \mathbf 2$, where $\struct{\mathbf 2, \vee, \wedge, \neg, \preceq}$ denotes the (Boolean lattice) two.

Point As Meet-Irreducible Element

A point of $L$ is a meet-irreducible element $p \in L$ that is not equal to $\top$.

Point As Continuous Map

A point of $L$ is a continuous map $f: 2 \to \mathbf L$, where $\struct{\mathbf 2, \vee, \wedge, \neg, \preceq}$ denotes the (Boolean lattice) two.

Also see

• Definition:Spectrum of Locale