Properties of $A + B$ related to $A$ and $B$. (GS2010)

Question

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 $\mathbb{R}$?

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

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

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

4. $A + B$ is connected when $A$ and $B$ are connected

I have tried as follows.

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

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.

3. 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 $A + B$.

4. I have no idea about it.

Thank you for your help. If this question is discussed earlier please give the link.

Answer 1

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.