Let $X,Y,Z$ be topological spaces and $f:X \rightarrow Y,g:Y\rightarrow Z$ are both continuos and $g \circ f$ is a homeomorphism. Prove that if $g$ is injection then $f,g$ are both homeomorphism.
There are any ways to prove $f,g$ are bijection ? I have tried to prove it but I didn't find the solution.
$f(x)=y(y)$ implies $g(f(x))=g(f(y))$ which implies $x=y$. So $f$ is an injection. Let $y \in Y$ and consider $g(y)$. We can write this as $g(f(x))$ for some $x$. Since $g$ is an injection we get $y=f(x)$. Hence $f$ is a surjection. I will leave it to you to complete the proof by showing that $g$ is a surjection.
