closed set form on product topology

Question

$X$=$X_1$×$X_2$ : product topology

let $A$×$B$ is closed set in $X$. then $A$ is closed set in $X_1$ also $B$ is closed set in $X_2$.

Is it right?

Answer 1

It is correct: for every $A\subseteq X_{1}$ and $B\subseteq X_{2}$ you have $\overline{A \times B}= \overline{A} \times \overline{B}$, and the result follows from thar fact.