A classical affine open subset and its coordinate ring

An affine open subset $U$ of a classical affine variety $X$ is an open subset equipped with an isomorphism, in the locally regular sense, to a classical affine variety. Assume AC for the coordinate-ring and principal-open interfaces used here. Its coordinate algebra is ${\mathcal{O}}_X(U)$; any specified affine realization identifies this algebra with that realization’s coordinate ring. Two such identifications differ by the pullback of their transition isomorphism in Classical affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms, so the algebra of functions on $U$ is independent of the realization. Nonempty principal opens have this property by Every nonempty principal open is a classical affine variety. Arbitrary opens are not declared affine.

Source comparison: Milne, Algebraic Geometry, v6.10, §3h pp. 71–72. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.