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Math Help - Compact in R

  1. #1
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    Compact in R

    Hello!

    I was wondering if someone could clear something up for me..

    Is every non-empty, compact subset of R a closed interval in R?
    It seems to me that this is true. I know that such sets are closed an bounded, but I don't know how to show that they're connected.

    Any help is much appreciated!
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  2. #2
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    Re: Compact in R

    According to the Bolzano–Weierstrass theorem, a subset of \mathbb{R}^n is compact if and only if it is closed and bounded. So, the union of two disjoint finite closed intervals in \mathbb{R} is compact.
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  3. #3
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    Re: Compact in R

    Quote Originally Posted by DrNerj View Post
    Is every non-empty, compact subset of R a closed interval in R?

    [0,1]\cup [2,3] is a non-empty, compact subset of \mathbb{R}~.

    Any non-empty closed and bound subset of \mathbb{R} is compact.
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  4. #4
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    Re: Compact in R

    Thanks a lot!
    That really helped clear things up.
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