6
$\begingroup$

I need an example of an isometry, $f$, from a metric space $X \rightarrow X$ that is not surjective. My example was letting the metric space be the set of non-negative real numbers with d$(x,y) := \max(x,y) - \min(x,y)$ if $x \neq y$, and $0$ if $x = y$. Then my isometry would be $f(x) = x+1$, so $0$ would not be in the image of $f$.

Is this a valid example? I don't know whether this is a real metric space, I just came up with it on my own.

$\endgroup$
  • $\begingroup$ Welcome to MSE! See here for a tutorial on how to format your math. $\endgroup$ – mrp Oct 18 '15 at 14:46
4
$\begingroup$

Your metric $d$ is the same as the standard metric $d'(x,y) = |x - y|$ on $[0,\infty)$ and your example indeed works.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.