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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- Dec 3rd 2011, 06:52 PMwopashuiprove 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 - Dec 3rd 2011, 07:23 PMDrexel28Re: prove subset of R has empty interior
Clearly the converse is automatically out. Anything with dense complement is going to have empty interior in $\displaystyle \mathbb{R}$. Any uncountable sets immediately spring to mind? Now, think about it, if $\displaystyle X\subseteq\mathbb{R}$ doesn't have empty interior you can find some $\displaystyle (a,b)\subseteq X$. What's the cardinality of $\displaystyle (a,b)$?

- Dec 4th 2011, 07:37 PMwopashuiRe: prove subset of R has empty interior
cardinality of (a,b) is continunm, so this implies that X is uncontable, which is a contridiction?

- Dec 4th 2011, 07:43 PMDrexel28Re: prove subset of R has empty interior
- Dec 6th 2011, 03:15 PMwopashuiRe: 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 - Dec 11th 2011, 11:20 PMDrexel28Re: prove subset of R has empty interior