Topology of the Segre product vs. the product topology

Suppose $V$ and $W$ are both infinite. As you note, we may assume $V$ and $W$ are both irreducible curves, and we may similarly assume that $V$ and $W$ are both affine. Now just take any nonconstant regular function $f:V\to k$ and any nonconstant regular function $g:W\to k$, and consider the regular function $h(v,w)=f(v)-g(w)$ on $V\times W$. Since $V$ and $W$ are irreducible curves, the fibers of $f$ and $g$ are finite, and also all but finitely many of the fibers must be nonempty. This means that if $C\subset V\times W$ is the vanishing set of $h$, then $C$ is infinite, but the two projections $C\to V$ and $C\to W$ both have finite fibers. No such set can be closed in the product topology (any infinite closed set in the product topology must contain a set of the form $\{v\}\times W$ or a set of the form $V\times\{w\}$).