Corollary 1

For every $x\in X$, let $I_x$ be the set of $f\in \mathscr{C}_0(X)$ vanishing at $x$. Then $x\mapsto I_x$ is a bijection of X onto the set of maximal closed ideals of $\mathscr{C}_0(X)$. These ideals are regular.

This follows at once from Prop. 1.

Let us denote by $X'$ the Alexandroff compactification of X, that is, the compact space deduced from X by adjunction of a point at infinity $\omega_X$ (TG, I, p. 67 and 68). The algebra $\mathscr{C}_0(X)$ is identified with the Banach algebra of continuous complex-valued functions on $X'$ vanishing at $\omega_X$.

For every $x\in X'$, we denote by ev$_x$ the character of $\mathscr{C}_0(X)$ defined by ev$_x(f) =f(x)$ for every $f\in \mathscr{C}_0(X)$.