 Ringed space

A ringed space $(X,{\mathcal{O}}_X)$ is a topological space together with a sheaf of rings describing its local functions. For an open subset $U$, ${\mathcal{O}}_X(U)$ is the ring of sections over $U$, with compatible restrictions and gluing. A smooth manifold with its real-valued smooth functions is a ringed space; a complex manifold with holomorphic functions is another. The underlying topology alone does not distinguish these kinds of functions. A locally ringed space additionally requires every stalk to be a local ring, which is the condition used in the definition of a scheme.

In the context of topology, a **ringed space** is a mathematical structure that consists of a topological space along with a sheaf of rings defined over that space. More formally, a ringed space is defined as a pair \( (X, \mathcal{O}_X) \), where: 1. \( X \) is a topological space. 2. \( \mathcal{O}_X \) is a sheaf of rings on \( X \).