The opposite Ring op of the category Ring of commutative rings with unit is an extensive category. Let R be a commutative ring. The extensive topology of any coherent analytic category is Stone, thus coherent. So the Pierce topology and extensive topology of a coherent analytic category are the same. Consider a set X as an object in the category Set of sets.
Any subset is a direct subset of X .
The extensive topology of X is simply the discrete space X .