# Thread: prove subset of R has empty interior

1. ## 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

2. ## Re: prove subset of R has empty interior

Originally Posted by wopashui
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 $(a,b)$?

3. ## 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?

4. ## Re: prove subset of R has empty interior

Originally Posted by wopashui
cardinality of (a,b) is continunm, so this implies that X is uncontable, which is a contridiction?
Right.

5. ## 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

6. ## Re: prove subset of R has empty interior

Originally Posted by wopashui
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.