# sum of two connected subset of $\mathbb{R}$

$A$ 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?

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Yes. $A\times B$ is a connected subset of $\Bbb R^2$, addition is continuous, and continuous maps preserve connectedness, so $A+B$ is connected.
Brian's proof is the rigorous one. For a non-rigorous picture proof, think of $A+B$ as translating the set $A$ over by the distances in $B$. The picture you get is a bunch of copies of $A$ sliding across the interval $B$ (it is a theorem that all connected sets in $\mathbb{R}$ are empty, singletons, or intervals, but the picture looks coolest when $B$ is an interval), and $A+B$ is the union of all these copies. This should convince you intuitively that it is connected.
This can be made rigorous: $A + B$ is the union of the connected sets $a + B$ for $a \in A$, and each $a + B$ intersects the connected set $A + b_0$, for some fixed $b_0 \in B$. (And $a + B$ and $A + b_0$ are clearly connected, being continuous images of $B$ and $A$, respectively.) – arkeet Dec 14 '12 at 7:41