Sheaf, étalé space with Riemann surfaces.

Question

Let $f:X\rightarrow Y$ be an holomorphic map betwen two Riemann surfaces and let:

$\Gamma:=${ $(x,y)\in X\times Y|y=f(x)$ } $\subset X\times Y$ be the graph of $f$. I have to show that $(\Gamma,proj_{|\Gamma},X) $, where $proj_{|X}:X\times Y\rightarrow X$ is the first projection, is an étalé space which is isomorphic to $X$. Thank you for any help.

Answer 1

Hints: Let $i_{X}$ denote the identity map on $X$. The map $i_{X} \times f:X \to X \times Y$ is a holomorphic bijection to $\Gamma$, and projection $\operatorname{proj}_{X}:X \times Y \to X$ on the first factor is a holomorphic map whose restriction to $\Gamma$ is the inverse of $i_{X} \times f$.