give an example of a complete metric space $(X,d)$ and a mapping $T: X \rightarrow X$ which does not have a fixed point in X and satisfies; $ d(T(x),T(y)) < d(x,y)$ $\forall x,y \in X, x\neq y$
i thought first that this was impossible by the fixed point theorem, but then figured with the final constraint it must be possible in some way. however i haven't been able to find any such example. I've tried with $\mathbb {R}$ and some subsets mostly, and used the discrete metric, but can't find a map to make this work...