I'm having trouble with this. Suppose that $U_1, U_2, \dots U_n$ are closed subsets of $\mathbb{R}^n$. I know that each $U_i' \subseteq U_i$ where $U_i'$ is the derived set but this doesn't feel like enough to show their cartesian product is closed.
Edit:
Suppose $S = U_1 \times U_2 \times \dots \times U_n$. Let $x \in \bar{S}$. Then $x \in \bar{U_i} = U_i$. Does this imply $x \in S$? I don't see why?
Best Answer
The projections $p_i$ onto the $i$-th coordinate are continuous.
$U_1 \times U_2 \times \ldots \times U_n = \cap_{i=1}^n (p_i)^{-1}[U_i]$ which is an intersection of closed sets so closed.