Finite fibres and an open immersion do not make a map finite

Assume the Axiom of Choice.

Given: AC, an algebraically closed field $k$, the affine line ${\mathbf{A}}_k^1$ with coordinate ring $k[t]$, the principal open $U=D(t)={\mathbf{A}}_k^1\setminus \{0\}$, and the inclusion $j\text{\hspace{0.17em}⁣}:U\hookrightarrow {\mathbf{A}}_k^1$.

$U=D(t)$ with its regular functions is an affine variety with coordinate ring $k[t,t^{-1}]=k[t{]}_t$, realized as the closed graph $\{ts=1\}\subseteq {\mathbf{A}}^1\times {\mathbf{A}}^1$, and $j$ is the restriction of the projection, with pullback the inclusion $k[t]\hookrightarrow k[t,t^{-1}]$ (Every nonempty principal open is a classical affine variety, Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms).

A morphism of classical varieties is quasi-finite when every closed-point fibre is a finite set; empty fibres are allowed (Quasi-finite classical morphisms).

A finite morphism of classical varieties is closed: the image of every closed subset is closed (Finite morphisms are closed with finite fibres).

The closed subsets of the affine line are the finite subsets and the whole line (On the affine line, the classical Zariski topology is cofinite); since $k$ is algebraically closed, hence infinite, the set ${\mathbf{A}}^1\setminus \{0\}$ is infinite and therefore not closed in ${\mathbf{A}}^1$. Its closure is all of ${\mathbf{A}}^1$.

AC is inherited through the classical localization, normalization, or finite-morphism suppliers cited above (The Axiom of Choice).