If $(M, d)$ is a metric space, $M$ is compact and $F: M\to M$ preserves distances (that is to say, $d(x,y)=d(F(x),F(y)) \ \forall x,y\in M$), then it must be proven that $F$ is an homeomorphism.
Because $M$ is compact and is Hausdorff (since it's a metric space), if $F$ is continuous and bijective then it's an homeomorphism.
I have already proven the continuity and injectivity from the preservation of distances, but I'm not sure how to prove exhaustivity.
