Assuming you give \(\displaystyle \mathbb{Q}\) the topology inherited from \(\displaystyle \mathbb{R}\) then open sets in \(\displaystyle \mathbb{Q}\) are those of the form \(\displaystyle A \cap \mathbb{Q}\) where \(\displaystyle A\) is open in \(\displaystyle \mathbb{R}\). Is \(\displaystyle A_1=\{ x \in \mathbb{R} : a<x<b \}\) open? \(\displaystyle A_1\) fails to be closed in \(\displaystyle \mathbb{R}\) because it lacks \(\displaystyle a\) and \(\displaystyle b\), but is that a problem in \(\displaystyle \mathbb{Q}\)?

Assuming you give \(\displaystyle \mathbb{Q}\) the topology inherited from \(\displaystyle \mathbb{R}\) then open sets in \(\displaystyle \mathbb{Q}\) are those of the form \(\displaystyle A \cap \mathbb{Q}\) where \(\displaystyle A\) is open in \(\displaystyle \mathbb{R}\). Is \(\displaystyle A_1=\{ x \in \mathbb{R} : a<x<b \}\) open? \(\displaystyle A_1\) fails to be closed in \(\displaystyle \mathbb{R}\) because it lacks \(\displaystyle a\) and \(\displaystyle b\), but is that a problem in \(\displaystyle \mathbb{Q}\)?