Let $X$ be a compact Hausdorff topological set, and $Y$ be its closed subset. Is the ideal of functions vanishing on $Y$ $$ I=\{f\in C(X):\ \forall y\in Y\ f(y)=0\} $$ complementable (as a closed subspace) in $C(X)$ (as a Banach space)?

This is true in the case when $X\subseteq {\mathbb R}^n$ (this follows from: Stein. Singular integrals... VI 2.2), but what about general case?

Pacific Journal of Mathematics1(1951), no. 3, 353--367. There are two nice papers, one due to Pełczyński and the second one due to Haydon, about spaces which satisfy the above-mentioned theorem. Google for the term: "Dugundji space". – Tomek Kania Aug 14 '14 at 13:42