How to find closed subsets of $A,B$ such that $\lambda A+(1-\lambda)B$ is not closed for a fixed $0<\lambda<1$.
As is well-known that if $A,B$ are bounded closed subsets of $\Bbb R^n$, then for any $0<\lambda<1$, $\lambda A+(1-\lambda)B$ is bounded and closed.
I'm suspecting that if $A,B$ at least one is unbounded, then $\lambda A+(1-\lambda)B$ need not to be closed. But I have no idea.
$A$ the $x$ axis, $B$ the $y$ axis, then $\lambda A+(1-\lambda)B$ is the whole $\Bbb R^2$, since $(x,y)=\lambda(x/\lambda,0)+(1-\lambda)(0,y/(1-\lambda))$. It is closed...