Complete Condition in Banach Fixed Point Theorem

Question

Can someone provide an example to show that for the Banach fixed point theorem, that is if $T : X → X$ is a contraction in a complete metric space $(X, d)$ then $T$ has a unique fixed point that $X$ is complete is an essential condition?

Answer 1

$$T \colon \mathbb{R}\setminus\{0\} \to \mathbb{R}\setminus\{0\};\; T(x) = x/2.$$

Basically, all examples are of that form, since a contraction is uniformly continuous, and can thus be extended to the completion of $X$, and the extension is also a contraction. So the extension $\widehat{T}\colon \widehat{X}\to\widehat{X}$ satisfies the premises of Banach's fixed point theorem, and has a unique fixed point.