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Math Help - Theorem about covers

  1. #1
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    Theorem about covers

    Theorem. Let D \subset Y. Suppose that D is non-empty and sequentially compact. Let C be an open cover of D. Then there is a real number \epsilon > 0 such that if E \subset D with d(E)<\epsilon then E \subset A_{\alpha} for some A_{\alpha}\in C.
    Note that d(E) is the diameter of E.

    This seems so intuitively obvious to me... or am I not reading it correctly? Can't I just choose \epsilon small enough that E is approximately a point?
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  2. #2
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    Re: Theorem about covers

    Quote Originally Posted by paupsers View Post
    Theorem. Let D \subset Y. Suppose that D is non-empty and sequentially compact. Let C be an open cover of D. Then there is a real number \epsilon > 0 such that if E \subset D with d(E)<\epsilon then E \subset A_{\alpha} for some A_{\alpha}\in C. Note that d(E) is the diameter of E.
    This seems so intuitively obvious to me... or am I not reading it correctly? Can't I just choose \epsilon small enough that E is approximately a point?
    This is far from a trivial result. It is known as proving that sequentially compact metric space has Lebesgue number.
    Suppose not. If n\in\mathbb{Z}^+ there must be a set B_n such that 0<d(B_n)<\frac{1}{n} but B_n is a subset of no open set in the cover.
    See what you can do with that setup.
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