For many quantum algebras A, by which we mean quantized coordinate rings, quantized Weyl algebras, and related algebras, good piecewise pictures of the prime and primitive spectra are known. More precisely, in generic cases there are finite stratifications of these spectra, based on a rational action of an algebraic torus, such that each stratum is homeomorphic to the prime or primitive spectrum of a commutative Laurent polynomial ring. What is lacking is an understanding of how these strata are combined topologically, i.e., of the Zariski topologies on the full spaces spec A and prim A. We develop a framework for the needed additional data, in terms of maps between the collections of closed sets of different strata, together with a conjecture stating how these maps should arise from homomorphisms between certain central subalgebras of localizations of factor algebras of A.