$A$ and $B$ are two connected subset of $\mathbb{R}$ define $A+B=\{x+y:x\in A,y\in B\}$, then is $A+B$ also connected?
naturally I was thinking about two disjoint connected subsets of $\mathbb{R}$, say $A=[0,1]$ and $B=[4,5]$
then $A+B=[4,6]$? so is it ingeneral true?