Does addition have to be defined in metric spaces? I have some questions, my first question is:
1. Does addition have to be dfined in a matric space?
I have some more quick questions, I will describe why I have them. Look at this excercise.

I wanted to know if I could generalize the excercise so that we do not work with R, but X-> Y and arbitary metric space Y.
I discussed this with the student-helper, and he said it could be done, but it was easier when we had R. But it seems like if you could generalize the excercise, then in all metric spaces, you would have to be able to add and multiply the objects? The student-helper said that this was taken care of in group and ring theory, and that we didn't have to worry about it in real analysis. He said all metric spaces had addition. Sadly I do not take group or ring theory.
But is he correct?
2. Is it correct that all metric-spaces must have addition and multipliction?
3. And if so, which of a), b) and c) can be generalized from x->R to an arbitary space Y, so that we get f,g: X->Y?
 A: Any set can be considered as a metric space with the discrete metric:
$$
d(x,y)=\begin{cases}
0 & \text{if $x=y$},\\
1 & \text{if $x\ne y$}.
\end{cases}
$$
What you can do as a generalization can be stated as follows. Assume $Y$ is a metric space and that $*$ is an operation on $Y$, that is, a map $Y\times Y\to Y$. If this map is continuous (with respect to the product metric on the domain), then you can easily prove that, when you define
$$
f*g\mapsto x \to f(x)*f(y)
$$
for $f\colon X\to Y$ and $g\colon X\to Y$, then $f*g$ is continuous.
There are other useful generalizations; if $Y$ is a Banach space (over $\mathbb{R}$, for instance), then addition in $Y$ and also the scalar multiplication $\mathbb{R}\times Y\to Y$ are continuous, so you can extend the result about maps $X\to \mathbb{R}$ to maps $X\to Y$.
However, in general, metric spaces don't need to have operations defined on them.
A: Answer to (1) and (2) (and thus also (3)) is no. Consider a finite discrete space with the discrete metric. This is a metric space, but has no addition or multiplication.
