The 2nd Ed. of Gamelin & Greene's Intro. to Topology contains as exercise 2.7 (c), p15, show the Cantor set is uncountable. A solution offered, on p198 relies on exercise 2.3, which states: "The set of isolated points of a countable complete metric space X forms a dense subset of X." The theory of the proof that C (Cantor Set) is uncountable is, as I understand it, is that since there is a dense subset of C and "none of these is isolated", C is uncountable. The complete metric space here is C. The dense subset of C are the endpoints of the middle-third intervals. The clincher is, since these are not isolated, C has no isolated points, and "by exercise 2.3 is uncountable."
In applying exercise 2.3, C is X and the set of middle-third endpoints is dense in C. The argument seems to be that if C were countable, its dense subset would have to have isolated points. My reading of 2.3 is if C had a set of isolated point and C were countable, then that set would be dense in C. I don't think that is the same as arguing that since the dense set has no isolated points, C is uncountable.
Comments please on the logic of this proof.
Your interpretation of the result in Exercise $2.3$ is incorrect. It’s actually a very strong result: it says that if $X$ is any countable, complete metric space, then $X$ has a dense set of isolated points. Thus, if $C$ were countable, it would be a countable, complete metric space and would therefore have a dense set of isolated points. In fact $C$ has no isolated points, so it certainly does not have a dense set of isolated points. Thus, it cannot be countable.