A short proof of the converse to the contraction principle and some related results

We simplify a proof of Bessaga's theorem given in the monograph of Deimling. Moreover, our argument let us also obtain the following result. Let $F$ be a selfmap of an arbitrary set $\Omega$ and $\alpha\in (0,1)$. Then $F$ is an $\alpha$-similarity with respect to some complete metric $d$ for $\Omega$ (that is, $d(Fx,Fy)=\alpha d(x,y)$ for all $x,y\in\Omega$) if and only if $F$ is injective and $F$ has a unique fixed point. Finally we present that the converse to the Contraction Principle for bounded spaces is independent of the Axiom of Choice.