Prove that the product topology on $X \times Y$ need not be the co-finite topology.

Question

Setup: Suppose that $X$ and $Y$ are topological spaces with the co-finite topology. Show that the product topology on $X\times Y$ need not be the co-finite topology.

Recall that the co-finite topology on $X$ is $\tau_x=\{\emptyset , A\subseteq X$ such that $A^c$ is finite$\}$. We also define the product topology as $\{\bigcup U\times V$ $|$ $U\in \tau_x,V\in\tau_y\}$.

Thoughts: I fail to see how the product topology need not be finite. For any example I think up, I keep trying to find some union of co-finite sets whose complement is infinite, but I can't seem to drum it up. Any hints are appreciated.

Answer 1

Consider $X=Y=\mathbb{R}$ and $A=B=(-\infty,0)\cup (0,\infty).$ We have that

$$(A\times B)^c=\{(x,y)\in\mathbb{R}^2|xy=0\}$$ is not finite.

Answer 2

If $X$ and $Y$ are infinite and $y \in Y$, then $X\times\{y\}$ is an
infinite proper closed subset. Thus $X\times Y$ is not cofinite.