You've got it exactly though. I'm taking this right from Rudin's
book, ironically enough. Any set with no limit points is closed
because it contains all of its limit points(0). This is why any
finite point set is closed(A corollary in Chapter 2 of Rudin). I
would reccommend re-reading Chapter 2 of Rudin's book for a further
explanation.
I believe you are correct that a countable set can not be compact.
Joel Gluskin
Wash U Academic Team, President
--- In quizbowl_at_yahoogroups.com, grapesmoker <no_reply_at_y...> wrote:
> And it seems we've lost the tack of the original remark in the
> process. The point made was that "compact" was ruled out by the use
> of "countable" so what I said in my previous post is wrong in that
> there are probably countable sets that are closed, but apparently
> not ones that are compact. Serves me right for not reading the post
> properly. I still stand by my statement that I don't believe in
> the "vacuously closed" proof, however.