$A+B$ closed if $A$ and $B$ are closed

Question

Let $A,B$ be subset of $\mathbb R^n$, both closed. I want to prove that $A+B$ is closed too. I know how to prove with one of both sets is compact but not without. Please help me, i struggle finding a solution.

This question is a duplicate, please dont answer no more.

False even in $\mathbb R$ and this has been answered many times. — Kavi Rama Murthy, Sep 04 '24 at 07:56
A counterexample is $A=\mathbb Z$ and $B$ the set of positive rational numbers $p/q$ in reduced form, with $p>q^2.$ Then $A+B$ is all the rational numbers. — Thomas Andrews, Sep 04 '24 at 08:03
@DimitryPetruschka Don't kick yourself. Asking questions is a big part of how we learn. — Thomas Andrews, Sep 04 '24 at 08:10
Hello new User! Please youse Mathjax, when typing down your questions to increase e.g. readability! Here is a link for a Tutorial: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference — Tina, Sep 04 '24 at 08:12

score 1 · Accepted Answer · answered Sep 04 '24 at 08:04

1

Define $A,B\subseteq\mathbb{R}^2$ by $A=\{(0,x)\mid x\leq -1\}$ and $B=\{(x,\frac{1}{x})\mid 0<x\leq 1\}$. Then $A,B$ are closed and $(0,0)\notin A+B$ but $(0,0)$ is in the closure of $A+B$ since $(x,0)\in A+B$ for all $0<x\leq 1$.

answered Sep 04 '24 at 08:04

dialegou

872

8

Please avoid answering duplicates. – Kavi Rama Murthy Sep 04 '24 at 08:05
Oh thank you, i see. – Dimitry Petruschka Sep 04 '24 at 08:06
2

@geetha290krm While it is true people shouldn't answer duplicate questions, until you find a duplicate, people don't know. – Thomas Andrews Sep 04 '24 at 08:08

$A+B$ closed if $A$ and $B$ are closed

1 Answers1