Homeomorphism between locally compact space $\Omega$ and maximal ideals space of $C_0(\Omega)$

Question

the following is a proposition:

If $\Omega$ is locally compact and $\Sigma$ is the maximal ideal space of $C_0(\Omega)$, then the map $x\to \delta_x$ is a homeomorphism.

To prove it, the author shows that the function $\phi:\Omega\to \Sigma$ is a bijection and open, and then concludes that it's a homeomorphism. But I do not know how does he conclude it. Please give me a hint.

Answer 1

Indeed, each continuous bijection from a compact topological space into a Hausdorff space is homeomorphism(See topological books). Similarly for open functions.