I am reading Springer's Invariant Theory. I already have some experience with linear algebraic groups and invariant theory, yet one of the first exercises of the book has already confused me.
In the first chapter, the book defines a linear algebraic group as a closed subgroup of some $GL(V)$, for some vector space $V$. Here, $V$ is a finite dimensional vector space over an algebraically closed field, $k$. Moreover, the topology on $V$ is the Zariski topology, so closed subgroup means Zariski closed.
Immediately after this definition, the book says the maps $g\mapsto g^{-1}$ and (for fixed $h\in G$) $g\mapsto hg$ or $g\mapsto gh$ are homeomorphisms and warns that:
But the product map $G\times G\rightarrow G$ need not be continuous.
which follows by the exercise
Prove that the product map $GL_1(k)\times GL_1(k)\rightarrow GL_1(k)$ is not continuous.
However, this product map is clearly continuous! $GL_1(k)=k\setminus\{0\}$ as a variety; thus, every proper closed subset is finite. For a point $x\in k\setminus\{0\}$, the inverse image of $x$ equals $\{ (\lambda,\mu)\in GL_1(k)\times GL_1(k)\mid \lambda\mu=x\}$ which is a closed set in $GL_1(k)\times GL_1(k)$. Therefore, the inverse image of every closed subset is also closed.
What am I missing?