Give an example of two closed sets $A, B \subseteq \mathbb{R}$ such that the set $A + B $ is not closed

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Possible Duplicate:
Sum of two closed sets in $\mathbb R$ is closed?

Give an example of two closed sets $A, B \subseteq \mathbb{R}$ such that the set $A + B = \{a + b : a \in A, b \in B\}$ is not closed.

This question appears on an old analysis qual I am studying. I know that both $A, B$ must be unbounded sets, because in an earlier part of the problem I have proved that $A + B$ is closed if either of the two sets are compact. The simplest unbounded and closed subset of $\mathbb{R}$ that I know is $\mathbb{Z}$. So I was starting with $A = \mathbb{Z}$, but I'm not yet able to come up with an appropriate $B$.

Hints or solutions are greatly appreciated.

I think this a dupe. But I am unable to find the duplicate. – Dec 30 '12 at 18:33 — , Dec 30 '12 at 18:33
@DavidMitra Good to know that my memory is good. – Dec 30 '12 at 18:41 — , Dec 30 '12 at 18:41

score 7 · Answer 1 · answered Dec 30 '12 at 18:36

7

Try $\mathbb Z+\sqrt 2\mathbb Z$.

answered Dec 30 '12 at 18:36

Hagen von Eitzen

374,180

Give an example of two closed sets $A, B \subseteq \mathbb{R}$ such that the set $A + B $ is not closed

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