The Mayer-Vietoris sequence is a powerful tool in algebraic topology that provides a way to compute the homology and cohomology groups of a topological space by breaking it down into simpler pieces. This sequence arises when a topological space can be decomposed into two open sets whose intersection is also open, allowing for a systematic way to relate the homology groups of the individual pieces to the whole space. It serves as a bridge connecting local properties of spaces to global topological features.
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