Continuous function on Hausdorff space

Question

I'm wondering about how to solve this one:

Let $X$ be a topological Hausdorff space and $A$ a subspace of $X$.

Let $f:X \rightarrow A$ continuous such that $f(a) = a \hspace{5mm} \forall a \in A$.

Show that $A$ is closed.

Any ideas? I think I might get the solution just by adding compactness to hypoteses..

Answer 1

Hint: Consider the map $g := (f, \operatorname{id}) : X \to X \times X$. If $\Delta_X$ is the diagonal in $X \times X$ (which is closed by the hypothesis that $X$ is Hausdorff) then what can you say about $g^{-1}(\Delta_X)$?