Let $X$ and $Y$ be two metric spaces with distances respectively $d_X$, $d_Y$. Is it possible to make contractions set a metric space?
Contracts set is the set of function from $X$ to $Y$ such that $d_X(x, y) \leq d_Y(f (x), f (y))$. I thought about using induced metric but I can't understand how to go on. Have you any hint?
I think you can consider the following distance $\delta$ on your set :
$$\forall f,g, \quad \delta(f,g)=\min \lbrace \sup_{x \in X}d_Y(f(x),g(x)), 1 \rbrace$$