The corona problem for planar open sets D and the of the maximal ideal space of H°°(D) are discussed and shown to depend only on the local behavior of D. Let D be an open subset of the Riemann sphere C*, and let H^iD) be the uniform algebra of bounded analytic functions on D. We will assume always that H^iD) contains a nonconstant function, that is, that C*\D has positive analytic capacity. Our object is to study the maximal ideal space ΛT(D) of H°°(D), and the fibers ^&(D) of ^£{Ώ) over points XedD. The basis for our investigation is the observation that the fiber ^tχ(D) depends only on the behavior of D near λ. This localization principle is used to obtain information related to the corona problem. The corona of D is the part of ^£φ) which does not lie in the closure of D. Our main positive results are that D has no corona under either of the following assumptions: (1) that the diameters of the components of C*\D (in the spherical metric, if D is unbounded) be bounded away from zero; or (2) that for some fixed m ^ 0, the complement of each component of D has ίgm components. The proofs rest on the localization principle, and on Carleson's solution of the corona problem for the open unit disc [2]. Each of the above conditions includes the extension of Carleson's theorem to finitely connected planar domains due to Stout [9]. In the negative direction, we present an example, due to E. Bishop, of a connected one-dimensional analytic variety W which is not dense in the maximal space of H°°(W). The construction is similar to that of Rosay [8].