Failure of rational approximation on some cantor type sets

Let A(K) be the algebra of continuous functions on a compact set K ⊂ ℂ which are analytic on the interior of K,and let R(K)be the closure (with respect to uniform convergence on K) of the functions that are analytic on a neighborhood of K. A counterexample of a question posed by A. O'Farrell about the equality of the algebras R(K)and A(K)when K = (K1 X [0, 1]) ∪ ([0,1] X K2) ⊆ ℂ,with K1 and K2 compact subsets of [0,1] is given. Also, the equality is proved with the assumption that K 1 has no interior. © 2008 American Mathematical Society.