Ringed space

Summary

A ringed space is a topological space together with a sheaf of commutative rings on it. The rings play the role of the rings of functions on the open sets. Manifolds with their smooth functions, complex manifolds with their holomorphic functions, and varieties with their regular functions are all ringed spaces. Schemes are defined as ringed spaces of a special kind, so the notion is the starting point of modern algebraic geometry.