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Math Help - What are the conected components of Q with topology from R?

  1. #1
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    What are the conected components of Q with topology from R?

    What are the conected components of Q with topology from \Re?

    I can prove that the inteior of Q in \Re is the empty set. I think that any subset A of \Re such that the interior of A is the empty set has connected components that are te singletons, but cannot prove this. Can anyone verify that this idea is correct please? Thanks.
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  2. #2
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    Quote Originally Posted by HenryB View Post
    What are the conected components of Q with topology from \Re?
    I think components are the singletons
    That is correct. If a connected set had two points, r < s, then there is an irrational number \gamma between r\;\&\; s.
    Think about two open sets \left( { - \infty ,\gamma } \right) \cap \mathbb{Q}\;\& \,\left( {\gamma ,\infty } \right) \cap \mathbb{Q}.
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    Yes, thanks. Was it not true then that any subset A of such that the interior of A is the empty set has connected components that are the singletons? Just out of curiosity. I've had a go at proving this but didn't quite get the result.
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    Quote Originally Posted by HenryB View Post
    (is) it not true then that any subset A of such that the interior of A is the empty set has connected components that are the singletons?
    I guess that I just don't understand your confusion.
    If the Int(A) = \emptyset then between any two points of A there is a point not in A.
    But that is exactly the idea I made in the posting above.
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