  The results in the paper are related to the classification problem for
invariant subspaces of multiplication operators in several variables. The main
results consist of characterizations, in the two dimensional case, of ideals of
polynomials and analytic functions which are closed in the relative topology
induced by Bergman spaces on certain domains. This provides proofs of the
Bergman space analogues of a conjecture of R. G. Douglas and V. Paulsen.
