We defined an isometry to be a bijection f:X\rightarrow X' such that d'(f(x_1),f(x_2))=d(x_1,x_2) $\forall x_1,x_2\in X$. Show that any isometry is a homeomorphism.
So my definition of homeomorphism is that a function f:X\rightarrow X' is a homeomorphism if $f$ is a bijection and $f^{-1}$ is continuous. So I have to show that
(a) $f$ is continuous.
$\forall\epsilon>0$ pick $\delta=f^{-1}(\epsilon)$. Then it follows that d(x_1,x_2)<\delta\implies d'(f(x_1),f(x_2))<\epsilon.
(b) $f^{-1}$ is continuous. Is this just a reverse of (a)?