Re: Sectionals questions

• From: grapesmoker <no_reply_at_yahoogroups.com>
• Date: Tue, 11 Feb 2003 22:59:13 -0000
• Message-ID: <b2bv81+3d0r_at_...>

--- In quizbowl_at_yahoogroups.com, jagluski <no_reply_at_y...> wrote:
> You're mistaken.  Closed does not disqualify countable.  A closed 
set 
> is a set that contains all of its limit points.  A countable set 
is 
> one that can be put into a 1:1 correspondence with Q.
> 
> For example, the natural numbers as a subset of the reals are 
closed 
> and countable.  Countable, you won't argue with.  There are no 
limit 
> points for this set, so the set is vacuously closed.  Thus, we 
have 
> found a set that is closed and countable. 

Pardon, but what does it mean to be "vacuously closed?" I'm afraid 
my copy of Rudin's "Principles of Mathematical Analysis" doesn't 
have that definition, so possibly you have some other source for 
that term? In either case, your logic seems fundamentally flawed: 
you claim that the naturals have no limit points, which is true, and 
then claim that they are thus closed, since they contain all of 
their limit points (none). That's like claiming that if I don't have 
any money, I possess all of my money (that is, none of it), which I 
don't quite buy (pun intended). I'm sure it's not that hard to prove 
that closed sets ought to be uncountable, but I don't have either 
the time or the inclination to do it. Finally, I'm inclined to 
believe whatever David says on this subject, since he knows about 
these sorts of things (mildly speaking).

Jerry