1. The problem statement, all variables and given/known data
It might not be a real topology question, but it's an exercise question in the topology course I'm taking. The question is not too hard, but I'm mainly doubting about the terminology:

2. Relevant equations
N.A.

3. The attempt at a solution
I would think not, unless I'm misunderstanding the terminology. Take the rational numbers and the subset denoted in [itex]\mathbb R[/itex] as [itex][\sqrt{2},2] \cap \mathbb Q[/itex]. It is indeed convex in [itex]\mathbb Q[/itex], but it's not an interval, cause I can't write it as [itex][q_1,q_2][/itex] or [itex]]q_1,q_2][/itex] with [itex]q_i \in \mathbb Q[/itex], or is my notion of interval too narrow?

Well, [itex][-\sqrt{2},\sqrt{2}][/itex] as a subset of R is not the same set as [itex][-\sqrt{2}, \sqrt{2}][/itex] as a subset of Q. The first contains many points not in the second.