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.