0
$\begingroup$

Let $X$ be a metric space and $t:X \rightarrow X$ a map that preserves distances $d(t(x),t(y))=d(x,y)$. Give an example in which $t$ is not bijective.

I thought every map that preserves distances was isometric and thus bijective? In the previous question in my book I had to prove that $t$ was injective, so I know I should find a $t$ that's not surjective.

  • 0
    One standard example is the (right-)shift operator on $\ell^2$. – 2017-02-09
  • 0
    Every map that preserves distances is injective, but not necessarily bijective. – 2017-02-09

1 Answers 1

3

Consider $X=\mathbb{N}$ with the discrete metric and the map $f\colon \mathbb{N}\to\mathbb{N}$, $f(n)=n+1$.