Is $D$ closed in $\Bbb R^n$?

Question

Let $C$ be a non-empty closed subset of $\Bbb R^n$ and $C × D$ be a closed subset of $C × \Bbb R^n$. Can we say that $D$ is closed in $\Bbb R^n$?

Answer 1

It is enough if $C$ is a nonempty set; it need not be closed. Fix $c\in C$ and let $d_j \to d$ with $\{d_j\} \subset D$. Then $(c,d_j) \to (c,d)$. Since $C \times D$ is closed we get $(c,d) \in C\times D$ which implies $d \in D$. Hence $D$ is closed.