Let $X$ be a TVS and $A\subseteq X$. Then it is known that for any open set $B$ in $X$, the set $A+B$ is also open. In particular, the sum of two open sets is again open. In Rudin's book in Functional Analysis, he pointed out that $A+B$ is closed whenever $A$ is compact and $B$ is closed. Of course, it is known that if $A$ and $B$ are closed then $A+B$ may fail to be closed. Am I right to say that
$A\times B$ is compact
whenever $A$ is compact and $B$ is closed? Assuming that $A\times B$ is compact, then by continuity of vector addition, it follows that the image of $A\times B$ under vector addition, which is $A+B$, is compact in $X$. It is known that every compact subset of aHausdorff topological space is necessarily closed. The fact that TVS is always Hausdorff, the result follows.
My worry right now is whether the question I raised is true or not. Any hint or solution is very much appreciated.