# Thread: What are the conected components of Q with topology from R?

1. ## 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.

2. Originally Posted by HenryB
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 $\left( { - \infty ,\gamma } \right) \cap \mathbb{Q}\;\& \,\left( {\gamma ,\infty } \right) \cap \mathbb{Q}$.

3. 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.

4. Originally Posted by HenryB
(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.