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(a) Let {U, U} be an open covering of a space X such that UU is nonempty and path-connected, and each U; is simply

(a) Let {U, U} be an open covering of a space X such that UU is nonempty and path-connected, and each U; is simply connected. Show that X is simply connected. [Hint: Consider the open cov- ering {f-(U)} of I, and use the Lebesque number lemma.] (b) Prove that S is simply connected for n 2.

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