User:Leigh.Samphier/Topology/Definition:Spatial Locale/Definition 2

Definition

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

Then:

$L$ is called a spatial locale if and only if:

$\forall a, b \in L : a \npreceq b \implies \exists$ a frame homomorphism $p: L \to \mathbf 2 : \map p a = \top, \map p b = \bot$

where $\mathbf 2$ denotes the Boolean Latice $\struct{\mathbf 2, \lor, \land, \neg, \preceq}$.

Such a locale is said to have enough points.

Also see

• User:Leigh.Samphier/Topology/Definition:Spatial Locale/Definition 1

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to locales, $\S 1$: Frames and locales, Definition $1.5$