Union of metric spaces Is the union of two metric spaces a metric space?
I tried it but could't define a suitable metric on intersection. Can somebody help me to understand it?

This might be cheating, but you can equip any set $Z$ with the discrete metric
$$d: Z \times Z \to \mathbb{R}^+$$ 
which is defined by $d(x,y) =0$ if $x=y$ and $d(x,y) =1$ if $x \neq y$ and the set $Z$  will become a metric space.
