Locally ringed space

In mathematics, a ringed space can be equivalently thought of either

Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry. The point of view (b) is more amenable to generalization; one simply needs to cook up a different way of parametrizing rings (cf. ringed topos).

Formally, a ringed space (X, O_X) is a topological space X together with a sheaf of rings O_X on X. The sheaf O_X is called the structure sheaf of X.

A locally ringed space is a ringed space (X, O_X) such that all stalks of O_X are local rings (i.e. they have unique maximal ideals). Note that it is not required that O_X(U) be a local ring for every open set U. In fact, that is almost never going to be the case.

An arbitrary topological space X can be considered a locally ringed space by taking O_X to be the sheaf of real-valued (or complex-valued) continuous functions on open subsets of X (there may exist continuous functions over open subsets of X that are not the restriction of any continuous function over X). The stalk at a point x can be thought of as the set of all germs of continuous functions at x; this is a local ring with maximal ideal consisting of those germs whose value at x is 0.

If X is a manifold with some extra structure, we can also take the sheaf of differentiable, or complex-analytic functions. Both of these give rise to locally ringed spaces.

...
Wikipedia