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We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.doi:10.1090/s0002-9939-2011-11168-7 fatcat:vrjtnhrzlngg3ic6le5bbdnnne