let L be a bounded distributive lattice with dual space ( X := I p...

Lydia Carey

Lydia Carey

Answered

2022-06-27

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.

Answer & Explanation

Jaylee Dodson

Jaylee Dodson

Expert

2022-06-28Added 22 answers

Step 1
Consider 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 2
The 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 .

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