Definition:Spectrum of Locale/Completely Prime Filters

Definition

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

Let $\map {\operatorname{pt}} L$ denote the set of points as completely prime filters of $L$.

For each $a \in L$, let:

$\Sigma_a = \set{p \in \map {\operatorname{pt}} L : a \in p}$

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

• 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter $\text {II}$: Frames and locales. Spectra, $\S 4$ Spectra, Definition $4.1$