  A framework is developed to describe the Zariski topologies on the prime and
primitive spectra of a quantum algebra $A$ in terms of the (known) topologies
on strata of these spaces and maps between the collections of closed sets of
different strata. A conjecture is formulated, under which the desired maps
would arise from homomorphisms between certain central subalgebras of localized
factor algebras of $A$. When the conjecture holds, spec $A$ and prim $A$ are
then determined, as topological spaces, by a finite collection of (classical)
affine algebraic varieties and morphisms between them. The conjecture is
verified for ${\cal O}_q(GL_2(k))$, ${\cal O}_q(SL_3(k))$, and ${\cal
O}_q(M_2(k))$ when $q$ is a non-root of unity and the base field $k$ is
algebraically closed.
