The following is Exercise 3.13.5 of Conway's Functional Analysis:
Let $X$ be a compact set and let $Y$ be a closed subset of $X$. A simultaneous extension for $Y$ is a bounded linear map $T:C(Y)\to C(X)$ such that for each $g\in C(Y)$, $T(G)_|Y=g$ . Let $C_0(X \setminus Y)=\{f\in C(X) ; f_|Y=0\}$. Show that if there is a simultaneous extension for $Y$, then $C_0(X\setminus Y)$ is complemented in $C(X)$.
For this I define $E: C(X)\to C_0(X\setminus Y)$ such that $Ef = f-T(f_|Y)$. Now we can show $E$ is a projection so $\operatorname{ran} E = C_0(X\setminus Y)$ is complemented in $C(X)$. Is it correct?
Thanks so much