# If $A$ and $B$ are closed subsets of the set of real numbers, then is $A+B$ closed?

This question already has an answer here:

• Sum of two closed sets in $\mathbb R$ is closed?

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#### Solutions Collecting From Web of "If $A$ and $B$ are closed subsets of the set of real numbers, then is $A+B$ closed?"

The result is not true. Take $A=\{\pi+n+1\colon n\in\mathbb N\}$, and $B=\left \{-n-1+\frac1 {n+2}\colon n\in \mathbb N\right\}$. Both sets are closed, $\pi$ is a limit point of their sum, but is not on their sum.

If both $A, B$ are compact, so is their sum (Since $A+B$ is the image of $A\times B$ under the continuous function $(x, y)\mapsto x+y$). Can you see what happens when one set is compact and the other isn’t?

Let $A$ be the set of negative integers.

Let $B$ be the set of all $n+\frac{1}{2^n}$ where $n$ ranges over the positive integers.

Then $A$ and $B$ are closed.

But $A+B$ is not closed, since it contains numbers arbitrarily close to $0$ but does not contain $0$.