On a converse to Banach's Fixed Point Theorem

TL;DR

This paper investigates the properties of metric spaces with the Banach Fixed Point Property (BFPP), demonstrating that certain open or simple sets with BFPP must be closed, and providing minimal examples of non-closed sets with BFPP.

Contribution

It proves that open or certain definable subsets of 2^n with BFPP are necessarily closed, and constructs the simplest non-closed examples in 2 with BFPP, including non-measurable sets.

Findings

Open or 2_ ext{si} 2_ ext{de} sets with BFPP are closed.

Existence of 2_ ext{si} and 2_ ext{de} non-closed sets with BFPP in 2.

Non-measurable sets can have the BFPP.

Abstract

We say that a metric space $(X,d)$ possesses the \emph{Banach Fixed Point Property (BFPP)} if every contraction $f:X\to X$ has a fixed point. The Banach Fixed Point Theorem says that every complete metric space has the BFPP. However, E. Behrends pointed out \cite{Be1} that the converse implication does not hold; that is, the BFPP does not imply completeness, in particular, there is a non-closed subset of $\RR^2$ possessing the BFPP. He also asked \cite{Be2} if there is even an open example in $\RR^n$, and whether there is a 'nice' example in $\RR$. 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 $X\su\RR^n$ is open or $X\su\RR$ is simultaneously $F_\si$ and $G_\de$ and $X$ has the BFPP then $X$ is closed. Then we show that these results are optimal, as we give an $F_\si$ and also a $G_\de$ non-closed example in $\RR$ with the BFPP. We also show that a nonmeasurable set can have the BFPP. Our non-$G_\de$ examples provide metric spaces with the BFPP that cannot be remetrised by any compatible complete metric. All examples are in addition bounded.