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Math Help - topology question

  1. #1
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    topology question

    me and my friend got into a little bit of argument...

    the question asks to find all subsets of the real numbers that are both compact and path connected. i was considering the closed interval [a,b] where a,b are in the real numbers, but my friend was telling me it should be the set of real numbers, which is a subset of itself. but i know that for the subset to be compact it must be closed and bounded by the heine-borel theorem... so is [a,b] all of the subsets or are there more? thanks in advance!
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  2. #2
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    There are exactly eight types of connected proper subsets of the reals:
    (a,b),[a,b),[a,b],(a,b],[a,\infty ),(a,\infty ),(\infty ,b],(\infty ,b)
    One of those is compact. As a space \mathbb{R} is itself not compact.
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