On a converse to Banach's Fixed Point Theorem

We say that a metric space $(X,d)$ possesses the Banach Fixed Point Property (BFPP) if every contraction $f:X\to X$ has a fixed point. The Banach Fixed Point Theorem states that every complete metric space has the BFPP. However, E. Behrends pointed out in 2006 that the converse implication does not hold; that is, the BFPP does not imply completeness; in particular, there is a nonclosed subset of $\mathbb{R}^2$ possessing the BFPP. He also asked if there is even an open example in $\mathbb{R}^n$, and whether there is a `nice' example in $\mathbb{R}$. In this note we answer the first question in the negative, the second one in the affirmative, and determine the simplest such examples in the sense of descriptive set theoretic complexity.

Specifically, first we prove that if $X\subset\mathbb{R}^n$ is open or $X\subset\mathbb{R}$ is simultaneously $F_\sigma$ and $G_\delta$ and $X$ has the BFPP, then $X$ is closed. Then we show that these results are optimal, as we give an $F_\sigma$ and also a $G_\delta$ nonclosed example in $\mathbb{R}$ with the BFPP.

We also show that a nonmeasurable set can have the BFPP. Our non-$G_\delta$ examples provide metric spaces with the BFPP that cannot be remetrized by any compatible complete metric. All examples are in addition bounded.