---
tag: KNDXDE7K8M
kind: concept
name: Ringed space
msc: [14A15, 54B40]
sources:
  - url: https://en.wikipedia.org/wiki/Ringed_space
same_as:
  wikidata: Q168344
  wikipedia: 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.

## Explanations

### L1 human

Many kinds of geometry fix a space and say which functions on it are allowed: continuous, smooth, holomorphic or polynomial.
A ringed space packages exactly this: a space, and for each open set a ring of allowed functions on it.
For example, $\R^n$ with the smooth functions on each open set is a ringed space.
The point of view lets one compare different geometries in one framework, and it allows "functions" that are not functions on points at all, as in scheme theory.