Consider the following definition. It seems to be standard, although I have not found a place where it is actually spelled out in this way.

**Definition.** A pointed scheme is *local* if is contained in every nonempty closed subset of .

**Example.** If is a local ring, then is a local scheme. Indeed, is contained in every nonempty closed subset , because every strict ideal is contained in .

We prove that this is actually the only example.

**Lemma.** *Let be a local scheme. Then is affine, and is a local ring whose maximal ideal corresponds to the point .*

*Proof.* Let be an affine open neighbourhood of . Then the complement is a closed set not containing , hence . Thus, is affine. Let . Let be a maximal ideal of ; then . Since this contains , we must have , i.e. corresponds to the (necessarily unique) maximal ideal .