Replace "vacuously" with "trivially", if that makes you feel better--
or, say that closed sets are sets which aren't missing limit points.
Then, any set without limit points is closed, since it isn't missing
any of its limit points (e.g., a single point is a closed set).
The natural numbers give an example of a countable, closed set;
David's example of the set of reciprocals of natural numbers gives an
example of a countable set which is not closed. So, if the question
gave "all countable sets" without qualifiers as an example, that
seems
wrong. Someone look at the question text, we probably missed
something
in the countable sets clue.
-Seth
--- 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.