Let $f:I \rightarrow S^2$ be a continuous injective function.Does $I$ homeomorphic to $f(I)$?
$I=[0,1]$.$S^2$ is a 2-sphere.
I think it's right,but I don't know how to prove that $f(U)$ is a open subset in $f(I)$ if $U$ is a open subset in $I$.If it's wrong, I also cannot come up with counterexample...
$f(I)$ is compact and any bijective continuous map between compact Hausdorff spaces is a homeomorphism. Instead of proving that $f(U)$ is open for any open set $U$ you can prove the equivalent statement that $f(C)$ is closed for any closed set $C$. Use the fact that $C$ is compact, so $f(C)$ is compact.