Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$A$ 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?

share|cite|improve this question
up vote 12 down vote accepted

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.

share|cite|improve this answer

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.

share|cite|improve this answer
Whee! Intervals on skis! :-) – Brian M. Scott Dec 14 '12 at 7:38
@BrianM.Scott That made me laugh :) – Gyu Eun Lee Dec 14 '12 at 7:40
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.