# a problem on topology

I can't solve the following problem.

Let $A = \{(x; y) \in \mathbb{R}^2 \mid \max\{|x|, |y|\} \leq 1\}$ and $B = \{(0; y) \in \mathbb{R}^2 \mid y \in \mathbb{R}\}$. Show that the set $A + B = \{a + b \mid a \in A; b \in B\}$ is a closed subset of $\mathbb{R}^2$.

Could you draw this set if asked? What definition of "closed" are you using? If it's "closed sets have open complements," have you thought about the complement of $A+B$ yet? –  Kevin Carlson Sep 20 '12 at 12:00
HINT: Show that $A+B = \{ (x,y) \in \mathbb{R}^2 : |x|\leq 1\}$ then prove that this set is closed by applying the definition of closed point, on every point, or you can look at the complement, that is $\{(x,y)\in\mathbb{R}^2 : |x|>1 \}$ and show that you can find an open ball around every point of that set.