As is known, for any compact space $T$ the Banach space $C(T)$ of all continuous functions on $T$ has the approximation property (see e.g. Albrecht Pietsch, Operator ideals). Is the same true for the (Hausdorff) quotient spaces of $C(T)$?
Does every quotient space $C(T)/X$ of $C(T)$ (where $X$ is an arbitrary closed subspace in $C(T)$, not necessarily an ideal) has the approximation property?