Quantum systems described by parametrized Hamiltonians are studied in a general context. Within this context, the classification scheme of Avron-Seiler-Simon for non-degenerate energy bands is extended to cover general parameter spaces, while their sum rule is generalized to cover cases with degenerate bands as well. Additive topological quantum numbers are defined, and these are shown to be conserved in energy band “collisions”. The conservation laws dictate that when some invariants are non-vanishing, no energy gap can develop in a set of degenerate bands. This gives rise to a series of splitting rules.