The punctured affine line is a principal open with Laurent-polynomial coordinate ring but is not closed in its ambient affine line

Assume the Axiom of Choice, and let $k$ be an algebraically closed field. Inside ${\mathbf{A}}_k^1$ with coordinate $t$, the punctured affine line is ${\mathbf{A}}_k^1\setminus \{0\}=D_{{\mathbf{A}}_k^1}(t)\mathrm{.}$ Therefore Regular functions on a principal open are the principal localization of the coordinate ring identifies its regular-function ring with the principal localization $k[t{]}_t,$ the Laurent polynomial ring obtained by adjoining $t^{-1}$.

By On the affine line, the classical Zariski topology is cofinite, the closed subsets of ${\mathbf{A}}_k^1$ are exactly the finite sets and the whole line. Hence $\{0\}$ is closed, so its complement ${\mathbf{A}}_k^1\setminus \{0\}$ is open; but that complement is not all of ${\mathbf{A}}_k^1$ and is infinite, so it is not closed. This is the basic quasi-affine example: it is affine as a principal open, but not as a closed subset of the ambient affine line.