Let Ω be an open subset of the
complex plane. Denote by 𝒪(Ω) the algebra of all holomorphic functions on Ω,
equipped with the topology of uniform convergence on compact set. The object of
this paper is to provide a complete classification of all the closed subalgebras of 𝒪(Ω)
which contain the polynomials, and apply this classification to several concrete
problems, including localness of these algebras, continuity of homomorphisms, and
number of generators. It should be emphasized that no assumptions are made as to
the connectivity of Ω. In fact, in the cases of most interest, Ω will not be
connected and some of the connected components of Ω will not be simply
connected.
