Definition:Spectrum of Locale/Continuous Maps

Definition

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

Let $\map {\operatorname{pt}} L$ denote the set of points as continuous maps of $L$, that is:

$\map {\operatorname{pt}} L$ is the set of continuous maps $f : \mathbf 2 \to L$

where $\struct{\mathbf 2, \vee, \wedge, \neg, \preceq}$ denotes the (Boolean lattice) two.

For each $a \in L$, let:

$\Sigma_a = \set{f \in \map {\operatorname{pt}} L : \map {\loweradjoint f} a = \top}$

where $\loweradjoint f : L \to \mathbf 2$ denotes the frame homomorphism such that $f = \paren{\loweradjoint f}^{\operatorname{op}}$

The spectrum of $L$, denoted $\map {\operatorname{Sp}} L$, is the topological space $\struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$

Also see

• Topological Equivalence of Definitions of Spectrum of Locale

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to locales, $\S 1$ Frames and locales, Lemma $1.3$