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common boundary is X. A further generalization was found by J. W. Alexander, who established the Alexander duality between the reduced homology of a compact subset X of Rⁿ⁺¹ and the reduced cohomology of its complement. If X is an n-dimensional compact connected submanifold of Rⁿ⁺¹ (or Sⁿ⁺¹) without boundary, its complement

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