Ringed space

Remarks

Example: let $X$ be a topological space and let $\mathcal O_X(U)$ be the ring of continuous functions $U \to \R$; then $(X, \mathcal O_X)$ is a ringed space. The stalk at each point $x$ is a local ring, whose maximal ideal consists of the germs that vanish at $x$.

Example: an algebraic variety $X$ over $k$, with $\mathcal O_X(U)$ the ring of regular functions on $U$, is a ringed space.

Example: any topological space $X$ with the sheaf of locally constant functions to a commutative ring $A$ is a ringed space.

Non-example: let $\mathcal B(U)$ be the ring of bounded continuous functions on an open set $U \subseteq \R$. Then $(\R, \mathcal B)$ is not a ringed space, because $\mathcal B$ is not a sheaf: the sections $x \mapsto x$ on the intervals $(-m, m)$ do not glue.

A ringed space whose stalks are all local rings is a locally ringed space; schemes and varieties are of this kind.