以前も貼ったと思うが

https://en.wikipedia.org/wiki/Sheaf_(mathematics)
(抜粋)
Ringed spaces and locally ringed spaces
Main article: Ringed space

A pair ( X , O X ) consisting of a topological space X and a sheaf of rings on X is called a ringed space. Many types of spaces can be defined as certain types of ringed spaces. The sheaf O X is called the structure sheaf of the space.
A very common situation is when all the stalks of the structure sheaf are local rings, in which case the pair is called a locally ringed space. Here are examples of definitions made in this way:

An n-dimensional Ck manifold M is a locally ringed space whose structure sheaf is an R -algebra and is locally isomorphic to the sheaf of Ck real-valued functions on Rn.
A complex analytic space is a locally ringed space whose structure sheaf is a C -algebra and is locally isomorphic to the vanishing locus of a finite set of holomorphic functions together with the restriction (to the vanishing locus) of the sheaf of holomorphic functions on Cn for some n.
A scheme is a locally ringed space that is locally isomorphic to the spectrum of a ring.
A semialgebraic space is a locally ringed space that is locally isomorphic to a semialgebraic set in Euclidean space together with its sheaf of semialgebraic functions.