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

**HenryB** · February 25th 2009, 11:25 AM

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.

**Plato** · February 25th 2009, 01:16 PM

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 $\left( { - \infty ,\gamma } \right) \cap \mathbb{Q}\;\& \,\left( {\gamma ,\infty } \right) \cap \mathbb{Q}$ .

**HenryB** · February 25th 2009, 01:48 PM

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.

**Plato** · February 25th 2009, 02:57 PM

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.

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