How could we state that $A$ and $B$ are compact if $A\times B$ is compact in $M \times N$?

Question

Assume that the Cartesian product of two non-empty sets $A \subseteq M$ and $B\subseteq N$ is compact in $M \times N$. Which steps should I follow to prove that $A$ and $B$ are compact?

Answer 1

I think you are asking if whether $A\times B$ compact implies $A$ (and $B$) is compact. Note that $\pi:A\times B\to A$ defined by $\pi:(a,b)\mapsto a$ is continuous and surjective...