Let me attempt to end this discussion, out of boredom:
* There exist countable subsets of the reals which are not closed
(hence not compact either). The question was flawed.
* A subset of the reals is compact if and only if closed and bounded
(Heine-Borel). This should help you decide which examples of closed
sets are compact.
* Given the other information quoted here (i.e. excluding the phrase
"or countable set"), either closed or compact would be acceptable
answers. However, closed is defined as a set with open complement,
hence was uniquely identified elsewhere in the question, IIRC.
Can we forget about it now?
--- In quizbowl_at_yahoogroups.com, grapesmoker <no_reply_at_y...> wrote:
>
> > "6. Considered as subsets of the real line, this term describes
> any
> > single point or countable set, any interval that includes its
> > endpoints, and any finite union of such sets."
> >
> > So, the question is indeed flawed, as there do exist sets which
> are
> > countable but not closed.
>
> It seems as though including the words "single point" is an
> assurance that "closed" makes sense as an answer, although I'm not
> sure if there are other answers that make sense as well.