let L be a bounded distributive lattice with dual space ( X := I p ( L ) , ⊆ , τ ), then the clopen downsets of X are X a , a ∈ L.

Name: let L be a bounded distributive lattice with dual space ( X := I p ( L ) , ⊆ , τ ), then the clopen downsets of X are X a , a ∈ L.
Uploaded: 2022-02-23T17:22:57+00:00
Description: let L be a bounded distributive lattice with dual space ( X := I p ( L ) , ⊆ , τ ), then the clopen downsets of X are X a , a ∈ L.

Question

let L be a bounded distributive lattice with dual space   (  X  :=            I        p    (  L  )  ,  ⊆  ,  τ  ), then the clopen downsets of X are       X    a    ,  a  ∈  L.

Jaylee Dodson · Accepted Answer

Step 1Consider a prime ideal I in a clopen downset D. The downset generated by I is the intersection of all the sets       X          a       such that   a  ∉  I. Because D is compact, there is a finite subintersection which is a subset of D. A finite intersection of sets of the form       X          a       is a set of the form       X          a      ,  so for each I in D there is some a such that   I  ∈      X          a        ⊆  D. Step 2The union of all sets       X          a        ⊆  D is therefore equal to D. Compactness yields a finite subunion, and a finite union of sets of the form       X          a       is again a set of the form       X          a      .