 Sheaf of rings

A sheaf of rings assigns a ring $\mathcal{O}(U)$ to each open subset $U$, and restriction ring homomorphisms for inclusions, satisfying locality and gluing. Compatible sections on an open cover must glue uniquely to a section on its union. Real-valued continuous functions on a topological space form a sheaf of rings under pointwise addition and multiplication: functions agreeing on overlaps define exactly one continuous global function. A constant presheaf is not generally already a sheaf; locally constant functions provide the corresponding sheaf construction.