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Let A and B be subsets of R. Define C = {a + b : a belongs to A, b belongs to B}. Pick out the true statements. (a) C is closed if A and B are closed. (b) C is closed if A is closed and B is compact. (c) C is compact if A is closed and B is compact. i think a is not true but cant find any counter example. no idea for b and c |
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For the first problem, let $A$ be the set of negative integers. Let $B$ be the set of all numbers of the form $n+\frac{1}{2^n}$, where $n$ ranges over the positive integers. Then $A$ and $B$ are closed. The set $A+B$ contains numbers arbitrarily close to $0$, but does not contain $0$. So $A+B$ is not closed. There is a very easy counterexample for the third assertion. Let $A$ be the integers, and $B$ almost anything. |
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HINTS: (a) What if $A=\left\{n+\frac1{n+1}:n\in\Bbb Z^+\right\}$ and $B=\Bbb Z$? Is $0\in A+B$? Is $0$ a limit point of $A+B$? (b) Let $\langle a_n+b_n:n\in\Bbb N\rangle$ be a sequence in $A+B$ converging to some real number $x$, where each $a_n\in A$ and $b_n\in B$. $B$ is compact, so $\langle b_n:n\in\Bbb N\rangle$ has a convergent subsequence $\langle b_{n_k}:k\in\Bbb N\rangle$. Let $b=\lim_{k\in\Bbb N}b_{n_k}$; then $b\in B$, and $\langle a_{n_k}+b_{n_k}:k\in\Bbb N\rangle$ converges to $x$. Then $\lim_{k\in\Bbb N}a_{n_k}=x-b$. (Why?) Is this limit in $A$? (c) What if $A=\Bbb Z$ and $C=\{0\}$? |
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