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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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    https://brainmass.com/math/geometry-and-topology/lebesque-number-connectivity-69863

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

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