Category:Definitions/Spectra of Locales

This category contains definitions related to Spectra of Locales.

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

Spectrum As Completely Prime Filters

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 $\struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$

Spectrum As Frame Homomorphisms

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

$\map {\operatorname{pt}} L$ is the set of frame homomorphisms $h : L \to \mathbf 2$

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

For each $a \in L$, let:

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

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

Spectrum As Meet-Irreducible Elements

Let $\map {\operatorname{pt}} L$ denote the set of points as meet-irreducible elements of $L$.

For each $a \in L$, let:

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

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

Spectrum As Continuous Maps

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 $\struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$