TITLE: Finite Product of Closed Maps Need Not Be Closed
QUESTION [11 upvotes]: What is an example of a finite product of closed maps that is not itself a closed map?

REPLY [2 votes]: Perhaps of some interest it might be the following example showing that the product of identical closed maps need not to be closed. Let $f\colon\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=\max\{0,x\}$. Then $f$ is closed since $f[A]$ is either $A$ or $(A\cap[0,\infty))\cup\{0\}$. But $f\times f$ is not, as it maps $\{(x,y)\!:x\cdot y=-1\}$ to $(\{0\}\times(0,\infty))\cup((0,\infty)\times\{0\})$.