The Mayer-Vietoris sequence exists in various homology and cohomology theories. A classical method in algebraic topology is to consider the local cases first, and then, extend them to global cases via Mayer-Vietoris sequences. We introduce notations such as cdp presheaf, cds precosheaf, Mayer-Vietoris system to develop this method systematically. As applications, we generalize Poincare dulaity theorem, Kunneth formulas, Leray-Hirsch theorem and write out explicit blow-up formulas of cohomology. In particular, we prove that the blow-up formula given by Rao, S., Yang, S. and Yang, X.-D. is still an isomorphism in the noncompact case.
