Closed graph of a function

Question

I have difficulties to answer at that question:

Let $X$ be a Hausdorff and compact topological space, and let $Y$ be a topological space. Let $f:X→Y$ be such that $G(f) = \{(x,f(x))|x∈X\} $ is a compact subset of $X \times Y$. Show that f is continuous.

I tried using the restrictions of the projections to $G(f)$, but I am block. I don't know how to use the Hausdorff condition in such a problem!

Answer 1

You are on the right track. What you want is that the projection map from the graph to $X$ is a homeomorphism. The relevant theorem is that a continuous bijection from a (quasi-)compact space to a Hausdorff space is a homeomorphism.

Answer 2

To show $f$ is continuous, let $C$ be a closed subset of $Y$. Suppose $\{ x_n\} \in f^{-1}(C)$ and $x_n \to x$. Then $(x_n,f(x_n)) \in G(f)$, and this has a convergent subsequence $(x_{n_k},f({x_{n_k}}))$ with limit $(x,y) \in G(f)$, so $y=f(x) \in X$ and since $f({x_{n_k}})) \to y$, it follows that $y=f(x) \in C$. Thus $x \in f^{-1}(C)$. Therefore $f^{-1}(C)$ is closed, hence $f$ is continuous as preimage of closed sets under $f$ is closed.