16 questions linked to/from Isometries of $\mathbb{R}^n$
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### A isometric map in metric space is surjective? [duplicate]

Possible Duplicate: Isometries of $\mathbb{R}^n$ Let $X$ be a compact metric space and $f$ be an isometric map from $X$ to $X$. Prove $f$ is a surjective map.
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### Isometry in a compact space is surjective

Let's say we have a compact metric space $(X,d)$ and a function $f: X\to X$ satisfying $d(f(x),f(y)) = d(x,y)$, $x,y\in X$. How can we show that this function is sujective in a simple way.
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### Examples of topology preserving transforms

I was trying to think of a transformation in Euclidean space which is topology preserving but not affine. I fact I could not think of even a non affine isometric transform in Euclidean space. Here ...

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