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Lebesque Number and Connectivity

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Lemma. Let {Ui} be an open covering of the space X having the following properties:
(a) There exists a point x0 such that x0Ui for all i.
(b) Each Ui is simply connected.
(c) If i≠j, then Ui Uj is arcwise connected.
Then X is simply connected.

Prove the lemma using the following approach:
To prove that any loop f: IX based at x0 is trivial, first consider the open covering
{f-1(Ui)} of the compact metric space I and make use of the Lebesgue number of this covering.
We say is a Lebesgue number of a covering of a metric space X if the following condition holds: any subset of X of diameter < is contained in some set of the covering.

Restate the lemma for the following special cases:
(1) A covering by two open sets
(2) The sets {Ui} are linearly ordered by inclusion

Using the restated lemma for special case (1), prove that the unit n-sphere Sn, n ≥ 2, is simply connected.

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Solution Summary

Lebesque number and connectivity are investigated and details are provided in the solution. The solution is detailed and well presented.