Properties of $A + B$ related to $A$ and $B$. (GS2010) Let $A$ and $B$ be subsets or $\mathbb{R}$. Define $A + B = \{ a + b : a \in A, b \in B\}$. Then which of the followings are true and which are false? Why? Please give a proof for truth and a counterexample or false. Will the situation be different if we consider $[0, \infty)$ instead of $\mathbb{R}$?


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*$A + B$ is bounded when $A$ and $B$ are bounded.

*$A + B$ is open when $A$ and $B$ are open.

*$A + B$ is closed when $A$ and $B$ are closed.

*$A + B$ is connected when $A$ and $B$ are connected
I have tried as follows.


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*is true as $a < M_1$ and $b < M_2$ gives $a + b < M_1 + M_2$.

*For any element $a+b$ we shall get open nbds $A'$ and $B'$ s.t. $a \in A' \subset A$ and $b \in B' \subset B$. So we shall get a nbd $A' + B'$ of $a + b$ in $A + B$. So $A + B$ is open. I have doubt.

*If $A$ and $B$ are finite and closed so $A + B$. When at least one of $A$ or $B$ will be infinite $A + B$ will have a limit point. I am not sure if it will stay in $A + B$.

*I have no idea about it.
Thank you for your help. If this question is discussed earlier please give the link.

Let me omit 1. It is true. 
In 2, you need only one of A or B to be open. 
For example, if A is open, then $A+B = \cup_{b \in B} \{A+b\}$ but each $A+b$ is an open set. 
For 3, see Closed sum of sets. 
For 4, a connected subset of the real line is necessarily an interval, and you can prove that addition of two intervals gives us an interval. 
