This theorem is from Munkres Topology 2nd ed.:
In the last para, it is said that 'choose an element' $n$ of $D$. Why we do not need axiom of choice here? (Here $D$ may not be finite.)
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The fact that we can choose an $n\in D$ is just the fact that $D$ is assumed nonempty. This doesn't require choice. As a heuristic, you don't need the axioms of choice to choose a sock from a pair of socks (or one sock from each pair of finitely many pairs of socks), but if I give you an infinite collection of pairs, you need some form of the axiom of choice to select one from each pair simultaneously. You can grab one element from each set in a finite collection of sets from the axioms of set theory. If you want to grab one from infinitely many sets, you need a stronger axiom in general.