Talk:Countably Compact Metric Space is Compact/Proof 2

I don't understand this proof. For example, how is it true that "if a metric space is countably compact it is by definition (?) second-countable"? Also, what purpose does the set $\left\{ {x_i} \right\}$ serve in the proof? Could someone please explain? Abcxyz 16:19, 15 March 2012 (EDT)


 * By definition of second-countableness, that is: having a topology with a countable basis, which is what has just been proved above.


 * The purpose of $\left\{{x_i}\right\}$ is the example used to demonstrate that there exists a dense subset of $A$ which is countable.


 * Is there a problem with this? It is the proof as published in which, as is pointed out copiously on this website, is far from being error-free. --prime mover 16:27, 15 March 2012 (EDT)