I'm reading a proof of Urysohn's lemma. The author constructs a sequence of open sets indexed by dyadic numbers, i.e. numbers of the form $\frac{p}{2^n}$. The proof starts at the bottom of page 4.
I don't seem to fully understand the proof, otherwise I could probably answer this question myself:
Where does he use that the indices are dyadic? At my current level of understanding it appears to me that one could use rationals to index these sets.
Many thanks for your help once again!