Analysis: Advance Calculus?

Prove that X=[0,1] U (2, 4] is not a compact set.

Using the definition: a set S is compact if and only if every family of open cover S has a finite subcover.

I understand the problem. I just cant think of an example.

1 Answer

  • Anonymous
    1 decade ago
    Favorite Answer

    The whole trick is in the (2 part. Here is a cover that would suit you:

    A = (-1/2, 3/2)

    U_1 = (3, 5)

    U_2 = (2 + 1/2, 5)

    U_3 = (2 + 1/3, 5)

    U_4 = (2 + 1/4, 5)


    U_i = (2 + 1/i, 5)


    It isn't hard to see that this is an open cover. But it doesn't have any finite subcover. Really, let's suppose that there is a finite subcover of {A, U_1, U_2, ...}. It is clear that since the subcover is finite, there is such an i that U_i, U_(i+1) and all the next intervals U_k, k>=i, are not in the subcover. But then the subcover does not cover the point 2+1/i, for example. So, this is is not a subcover (a contradiction).

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