Intersection of open affines in a prevariety

Question

What is an example of a prevariety in which the intersection of some two open affines is not an open affine? My examples of prevarieties that are not varieties does not extend beyond the affine line with the origin doubled. My guess is that the plane with four origins might give an example but I don't see how. Thanks.

Answer 1

I think the plane with doubled origin is enough. From the construction you see two copies of $\mathbb{A}^2$ which intersect in a copy of $\mathbb{A}^2 - 0$.