Help with a proof regarding empty interior

Question

Show that every countable subset of $\mathbb{R}$ has empty interior in $\mathbb{R}$ and therefore is included in its own boundary ?

I have no idea how to proceed with this one.

Any help would be really appreciated.

I guess by $R$, you mean the set of all real numbers, it would have been better if you have described the topology. — Math137, Oct 10 '13 at 10:59

score 1 · Answer 1 · answered Oct 10 '13 at 11:08

1

Hint: Every set with a nonempty interior contains a nonempty open subset. Every nonempty open set contains an interval with positive radius. Every interval with positive radius...

answered Oct 10 '13 at 11:08

Did

279,727

score 0 · Answer 2 · answered Oct 10 '13 at 11:07

0

Suppose $X \subseteq \mathbb{R}$ has nonempty interior. Then it contains an open set, and in particular, it contains an open interval. So can it be countable?

answered Oct 10 '13 at 11:07

Clive Newstead

64,925

score 0 · Answer 3 · answered Oct 10 '13 at 11:08

0

Hint: The interior of a set is an open set. What can you say about open subsets of $\mathbb R$?

answered Oct 10 '13 at 11:08

Carsten S

8,726

Help with a proof regarding empty interior

3 Answers3