In the paper some aspects of the topological structure of the maximal ideal space \(M(H^\infty) \) of the uniform algebra \(H^\infty \) of bounded analytic functions on the open unit disk in the complex plane are investigated. The authors discuss properties of the cluster points of a sequence \((x_n)_{n} \) in \(M(H^\infty). \) In general it is assumed that the points \(x_n \) are in different fibers \(M_{\lambda_n}, \) the set of all \( m\in M(H^\infty) \) such that \( id (m)=\lambda_n \) where \(id \) is the identity function on the unit disk and \( \lambda_n \) are points on the unit circle such that the argument of \( \lambda_n \) tends monotonically to zero.
To give a flavor of the paper we mention two results: (1) If \( x_n \) has trivial Gleason part for each \( n\) then every cluster point of \((x_n)_{n} \) has a strictly maximal support set, and (2) if \( x_n \) has non-trivial Gleason part for each \(n \) then every cluster point of \((x_n)_{n} \) has a Gleason part which is homeomorphic to the open unit disk. The paper contains further interesting results along these lines.