# Proof of Urysohn's lemma — dyadic rationals

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!

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Yes, you can use rationals to index the sets, but some authors think the dyadic argument is slightly nicer to present. –  Qiaochu Yuan Apr 6 '11 at 18:47

He uses the dyadic rationals because they have a nice structure. All the ones with denominator $2^n$ are at height $n$ and between any two neighbors at height $n$ there is exactly one at height $n+1$. This allows him to use Lemma 3.7 to construct $U_r$ for each dyadic rational $r$ and have it interleave properly.