Question: (b) Give a precise statement of Leray's theorem about the nerve NU of an open cover U of a space X. For the sphere S?

(b) Give a precise statement of Leray's theorem about the nerve NU of an open cover U of a space X. For the sphere S? = {x E R3 : [(x| | = 1} consider the North Pole (0, 0, 1), South Pole (0, 0, -1) and open cover U = {U1, U2} with U1 = S2\\ { (0, 0, 1) }, U2 = $2\\ {(0, 0, -1) }. Decide if | NU| is homotopy equivalent to S2, and explain why your answer is consistent with Leray's theorem. [12 marks]

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