Q:

Does $A=\{x\in\mathbb{Q}: x>0\}$ have any isolated points?

Let $A=\{x\in\mathbb{Q}: x>0\}$ and $B=\{x\in\mathbb{R}: x>0\}$. I think that it is easy to understand and to prove that the set $B$ does not contain any isolated points, but can we say the same about set $A$? Are the point's in $A$ all isolated? 

A:

If $x<y \in A$ then $\frac{x+y}{2} \in A$ and $x<\frac{x+y}{2}<y$.Since $A$ contains at least two points, no point in $A$ can be an isolated point.

