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 $\set {x_i}$ 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 $\set {x_i}$ 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 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) which, as is pointed out copiously on this website, is far from being error-free. --prime mover 16:27, 15 March 2012 (EDT)

Where was it proved that the topology on a countably compact metric space has a countable basis?

As for the set $\set {x_i}$, it was just mentioned that a Sequentially Compact Metric Space is Separable, so isn't this redundant?

As for the question if there is a problem with this proof, I don't know. I just want to understand the argument. Abcxyz 16:45, 15 March 2012 (EDT)

dunno then --prime mover 16:47, 15 March 2012 (EDT)