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Let $\Omega$ be a circular domain, that is, an open disk with finitely manyclosed disjoint disks removed. Denote by $H^\infty(\Omega)$ the Banach algebraof all bounded holomorphic functions on $\Omega$, with pointwise operations andthe supremum norm. We show that the topological stable rank of$H^\infty(\Omega)$ is equal to 2. The proof is based on Suarez's theorem thatthe topological stable rank of $H^\infty(\D)$ is equal to 2, where $\D$ is theunit disk. We also show that for domains symmetric to the real axis, the Bassand topological stable ranks of the real symmetric algebra$H^\infty_\R(\Omega)$ are 2.

Topological Stable Rank of $H^\infty(\Omega)$ for Circular Domains $\Omega$