The long exact sequence of a pair is a fundamental concept in algebraic topology that connects the cohomology groups of a topological space and its subspaces. This sequence arises from a pair of spaces, typically denoted as $(X, A)$, where $X$ is a topological space and $A$ is a subspace of $X$. The long exact sequence provides a systematic way to understand how the cohomology groups of the space $X$ relate to those of the subspace $A$ and the quotient space $X/A$, revealing deeper properties about their topological structure.
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