{
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  "slug": "excision-theorem",
  "title": "Excision theorem",
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  "lead": "In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X and subspaces A and U such that U is also a subspace of A , the theorem says that under certain circumstances, we can cut out (excise) U from both spaces such that the relative homologies of the pairs (X \\setminus U,A \\setminus U ) into (X, A) are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.",
  "body": "In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X and subspaces A and U such that U is also a subspace of A , the theorem says that under certain circumstances, we can cut out (excise) U from both spaces such that the relative homologies of the pairs (X \\setminus U,A \\setminus U ) into (X, A) are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute. === Invariance of dimension=== If nonempty open sets U\\subset \\mathbb{R}^n and V\\subset \\mathbb{R}^m are homeomorphic, then m = n. This follows from the excision theorem, the long exact sequence for the pair (\\mathbb{R}^n,\\mathbb{R}^n-x) , and the fact that \\mathbb{R}^n-x deformation retracts onto a sphere. In particular, \\mathbb{R}^n is not homeomorphic to \\mathbb{R}^m if m\\neq n . ==References== ==Bibliography== Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, Allen Hatcher, Algebraic Topology. Cambridge University Press, Cambridge, 2002.",
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    "title": "Excision theorem",
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    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
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