EXERCISES

1. (15) For each of the following sets, find the interior, the closure, and the boundary. Then determine which of the sets are open, closed, neither, or both.
   1. 
      
      $$\left\{ (x_1,x_2) : x_1 \geq 0, x_2 \geq 0 \right\}.$$
      
      

   2. 
      
      $$\left\{ (x_1,x_2) : x_1 > 0, x_2 > 0 \right\}.$$
      
      

   3. 
      
      $$\left\{ (x_1,x_2) : x_1 > 0, x_2 \geq 0 \right\}.$$
      
      

   4. 
      
      $$\Re^n$$
      
      

   5. 
      
      $$\left\{ (x_1,x_2) : x_1^2 + x_2^2 < 0 \right\}.$$
      
      

   6. 
      
      $$\emptyset .$$
      
      

2. 1. (10) Prove that $D$ is closed if and only if the complement $D^c$ is open.
   2. (10) Prove that $x \in \partial D$ if and only if for any $r > 0$ there exists a $y \in B(x;r) \cap D$ and a $z \in B(x;r) \cap D^c.$