It is well known that there is a generalized Mayer Vietoris exact sequence in the context of smooth manifolds and the Rham cohomology, where a sequence involving countably many open sets is obtained. It is explained in Bott and Tu's book Differential Forms in Algebraic Topology.
Is there an analogous result for a covering by a finite family of open sets in the topological setting? Could you give me a reference?
What I exactly mean: I want a kind of Mayer-Vietoris exact sequence involving a covering of a compact topological space (such as a manifold (topological) or a finite CW-complex) by a finite family of open sets (not necessarily two sets). (Imagine I am working with locally trivial fibre bundles and Leray-Hirsch is not enough). In reference to the homology or cohomology theories involved, the singular ones would be nice. And I am working with coefficients in a field if that helps (so in the case of singular theories homology and cohomology are isomorphic).
EDIT: I want to avoid spectral sequences. Thanks in advance.