before I write the problem I just want to clarify that this is not a graded assignment but a practice problem designed to help understand the topic. My professor has limited office hours so I am hoping someone can help me reason this out.

does the set Q intersection [0, 1] in R have an open cover that cannot be replaced by a finite sub-cover?

this is what I know:

(1) the question is asking if Q intersection [0, 1] is compact in R

(2) that [0, 1] is a compact interval

(3) that Q intersection [0, 1] is contained in [0,1] and that by the density of Q the Q intersection [0, 1] will be an open set

my initial thought is that no this intersection is not compact but I am not confident in that answer and I would really appreciate any assistance.

Thank you