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Math Help - prove subset of R has empty interior

  1. #1
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    prove subset of R has empty interior

    Prove that any countable subset of R has empty interior. Is the converse true?

    this is one of the question in my midterm that i couldnt do, the prof are not giving out solution, need some help to prove this
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    MHF Contributor Drexel28's Avatar
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    Re: prove subset of R has empty interior

    Quote Originally Posted by wopashui View Post
    Prove that any countable subset of R has empty interior. Is the converse true?

    this is one of the question in my midterm that i couldnt do, the prof are not giving out solution, need some help to prove this
    Clearly the converse is automatically out. Anything with dense complement is going to have empty interior in \mathbb{R}. Any uncountable sets immediately spring to mind? Now, think about it, if X\subseteq\mathbb{R} doesn't have empty interior you can find some (a,b)\subseteq X. What's the cardinality of (a,b)?
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    Re: prove subset of R has empty interior

    cardinality of (a,b) is continunm, so this implies that X is uncontable, which is a contridiction?
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    MHF Contributor Drexel28's Avatar
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    Re: prove subset of R has empty interior

    Quote Originally Posted by wopashui View Post
    cardinality of (a,b) is continunm, so this implies that X is uncontable, which is a contridiction?
    Right.
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    Re: prove subset of R has empty interior

    what could be an explicit counter example for the converse?

    that is we need to find a subset C of R such that int(C)=empty but C is uncountable
    Last edited by wopashui; December 6th 2011 at 04:45 PM.
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    MHF Contributor Drexel28's Avatar
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    Re: prove subset of R has empty interior

    Quote Originally Posted by wopashui View Post
    what could be an explicit counter example for the converse?

    that is we need to find a subset C of R such that int(C)=empty but C is uncountable
    Editing long after your initial post is a bad idea, especially when the before-edit-post was conversation-ending. But, yes, that sounds like a good idea. Hint: what is ALWAYS the counterexample when dealing with dense things, be rational dear sir.
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