What assumptions must be made for a metric space $X$ such that all continuous functions $f: X\rightarrow X$ have a fixed point?

Question

So basically, I’m looking for a set of constraints on a metric space $X$ such that every continuous function from $ X\rightarrow X$ has a fixed point. Obviously such metric spaces do exist, with the trivial example being $X = \{x\}$ (equipped with any metric), but I would be surprised if that was the only example.

At first I thought maybe it would be enough for $X$ to be compact and complete, but the unit circle is a counterexample to that.

Answer 1

The fixed point theorem of Brouwer might be what you're looking for

Conditions: X is compact and convex (in an Hilbert space - it's also true for Banach spaces by Schauder fixed point theorem )

Answer 2

See this Wikipedia article on the Fixed Point Property of topological spaces.

Answer 3

The Lefschetz fixed point theorem is a generalization of Brouwer's fixed point theorem. See e.g.

en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem mit.edu/~ivogt/LefschetzFixedPointTheorem.pdf

It deals with compact polyhedra $X$ and gives a sufficient criterion in order that each $f : X \to X$ has a fixed point.