Contraction mapping with no fixed point using a incomplete metric space

Question

I know that if $f:X\rightarrow X$ is a contraction, then $d(f(x),f(y))\leq \alpha d(x,y)$ for $0<\alpha<1$.

I'm looking for a counter example, that is a metric space that's incomplete, and where there are contractions with no fixed point.

Can somebody give me an example of such?

Answer 1

Just take a contraction with a unique fixed point, and remove that point. For example $x \to x/2$ on $\mathbb{R} - \{0\}$ is a contraction with no fixed points.