TITLE: The Use of the Axiom of Choice in an Elementary Proof
QUESTION [20 upvotes]: I wanted to give some of the new undergrad analysis students the following problem: given the real numbers (with the standard topology, as they'd expect) one cannot have an uncountable set such that every point is an isolated point.  A few of my fellow grad students attempted solutions first.
A sketch of a potential proof was given as follows: take each point and create as large a ball as is possible without it containing any other point of the set, so say $(a-\alpha, a+\alpha)$.  Then construct corresponding balls with a smaller radius: $(a-\frac{\alpha}{2}, a+\frac{\alpha}{2})$.  These new balls will intersect each other trivially.  Then choose a rational point from each of these new balls.  This set will be in one-to-one correspondence with the balls which are in one-to-one correspondence with the points of the original set.
After presenting this proof, it was argued that it requires the (non-finite) axiom of choice (as we are picking one point from a potentially uncountable set).  We modified the proof by using the density of the rationals to pick rationals $p_{a},q_{a}$ such that $(a-p_{a},a+q_{a})\subseteq (a-\frac{\alpha}{2}, a+\frac{\alpha}{2})$, and then have our function pick the "left-most" end-point.  It was still argued that this used the axiom of choice.
Because I am not entirely familiar with what constitutes use of the axiom of choice, I wanted to open the question up to all of you.  Do these proofs require the (non-finite) axiom of choice?  If so, is there a proof you know of which does not require it?

REPLY [6 votes]: As I mentioned recently in another answer, this problem comes down to two simple facts:
1) A subspace of a topological space admitting a countable base (i.e., a second countable space) again admits a countable base: just intersect every element of a countable base with the subspace.
2) A discrete space has a countable base if and only if it is countable.  
In particular there is no choice going on here.
(I should note that the notion of "base for a topology" is more commonly encountered in a first topology course than an undergraduate real analysis course.  I can see some merits to the more explicit approach given in the question.  But for those who know the smallest (nonzero!) amount of general topology, this approach seems to me to be much cleaner and simpler.)