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Math Help - compact sets

  1. #1
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    compact sets

    before I write the problem I just want to clarify that this is not a graded assignment but a practice problem designed to help understand the topic. My professor has limited office hours so I am hoping someone can help me reason this out.


    does the set Q intersection [0, 1] in R have an open cover that cannot be replaced by a finite sub-cover?

    this is what I know:
    (1) the question is asking if Q intersection [0, 1] is compact in R
    (2) that [0, 1] is a compact interval
    (3) that Q intersection [0, 1] is contained in [0,1] and that by the density of Q the Q intersection [0, 1] will be an open set

    my initial thought is that no this intersection is not compact but I am not confident in that answer and I would really appreciate any assistance.

    Thank you
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: compact sets

    Quote Originally Posted by bskcase98 View Post
    I am not confident in that answer and I would really appreciate any assistance.
    Denote A=\mathbb{Q}\cap [0,1] then, \bar{A}=[0,1]\neq A, that is, A is not a closed set, as a consequence, A is not compact.
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