Let $X$ be a topological space and $Y$ compact Hasudorff space. Prove that a function $f:X\to Y$ is continuous iff the graph of $f$, $\Gamma_f$, is a closed subset of the product space $X \times Y$
I have down the $\Leftarrow$ direction. For $\Rightarrow$, i have the following idea so far:
We want to show $f$ is continuous, i.e. For $U\subseteq Y, \implies f^{-1}(U)\subseteq X$ is open.
Since $\Gamma_f$ is a closed subspace of $X\times Y$, by the product topology we know $\{x:x\in X\}=X\subseteq X$ is closed and $\{f(x):x\in X\}=f(X)\subseteq Y$ is closed. Since $f(X)$ is a closed subspace of a compact space, $f(X)$ is also compact, so for any open cover $\mathcal{U}_i=\cup_{i\in \mathcal{I}} U_i$ of $f(X)$, there exists a finite subcover, $\mathcal{U}_{i_n}$.
From here, we should map back to $X$ I think. So $f^{-1}(\mathcal{U_i})$ is a cover of $X$ and so is $f^{-1}(\mathcal{U_{i_n}})$. From this we want to show that any given $U_i$ is open. But I'm stuck.
Thanks for any help!
