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# Lebesque Number and Connectivity

Lemma. Let {Ui} be an open covering of the space X having the following properties:
(a) There exists a point x0 such that x0&#61646; Ui for all i.
(b) Each Ui is simply connected.
(c) If i&#8800;j, then Ui&#61639; Uj is arcwise connected.
Then X is simply connected.

Prove the lemma using the following approach:
To prove that any loop f: I&#61614;X 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 &#61541; is a Lebesgue number of a covering of a metric space X if the following condition holds: any subset of X of diameter < &#61541; 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 &#8805; 2, is simply connected.

#### Solution Summary

Lebesque Number and Connectivity are investigated. The solution is detailed and well presented.

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