In Petersen's Riemannian Geometry book I encountered the following statement : Any compact metric space $X$ is Borel equivalent to some $S \subset [0, 1]$ i.e. there is a bijection $f : X \rightarrow S$ which is measurable and whose inverse is also measurable. Frankly speaking I have no clue how to prove this. I know some theorem which says (if I remember correctly) $X$ is homeomorphic to a closed subset of $\{0,1\}^\mathbb{N}$. Is this of any use here? Can someone give some hint?
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1Section 7.1 of these notes is a nice exposition of a more general result. – May 24 '15 at 19:16
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@user147263 link is dead :( – Carlos Esparza Oct 30 '22 at 04:22