Let us consider a contraction mapping $f$ acting on metric space $(X,~\rho)$ ($f:X\to X$ and for any $x,y\in X:\rho(f(x),f(y))\leq k~\rho(x,y),~ 0 < k < 1$). If $X$ is complete, then there exists an unique fixed point. But is there an incomplete space for which this property holds as well? I think $X$ should be something like graph of $\text{sin}~{1\over x}$, but I don't know how to prove it.

Thank you and sorry for my english.