I read the proof that uses tube lemma and I do not have any problem with it but I cannot see what is wrong with the proof that first came to my mind:
Let $X$ and $Y$ be compact spaces. Let $\mathcal{A}$ be an open covering of $X\times Y$. Then, $\bigcup_{U\in \mathcal{A}}{U}=X\times Y$. And $\pi_1(\bigcup_{U\in\mathcal{A}}{U})=\bigcup_{U\in\mathcal{A}}{\pi_1(U)}=X$, similarly $\bigcup_{U\in\mathcal{A}}{\pi_2(U)}=Y$. Since $X$ is compact, there are finitely many $\pi_1(U)$'s and $\pi_2(U)$ that cover $X$ and $Y$. So the product of these finitely many sets covers $X\times Y$.
I presume that there is a mistake related to basic set theory.
Best Answer
Yes, there is a mistake in this proof. The problem is that when you have your finite covers of $X$ and $Y$, it is true that the product of those open sets cover $X \times Y$ ; what is not true is that those products were in your original open cover!
Hope that helps,