Consider an isometry from a compact metric space to itself, this is simply a function that preserves distances. Notice that this implies that the function is continuous and moreover that it is injective. What we have to prove is that such a function is also surjective. Observe that this implies that the function is indeed a homeomorphism.

A consequence of this fact is that the category that has compact metric spaces as objects and isometries as arrows enjoys a Cantor-Schröder-Bernstein-like property.

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HINT: the complement of the image of the map is an open set (why?)

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[…] seen in this post, any isometry from a compact metric space to itself is surjective, so the set of all such […]

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