Z and Baire Category Theorem

From: Tim Brauch (RnEeMwOs.pVoEst_at_tbrauch.cNOoSPAMm)
Date: 09/13/04 

Date: Mon, 13 Sep 2004 03:42:32 GMT

I am taking a course in Real Analysis, Measure and Integration to be more
specific. This area is far from my cup of tea, numerical and applied
linear algebra especially digital image processing. Thus, I am struggling
quite a bit.

I was able to prove the set of integers, Z, with the absolute value metric
is a complete metric space: any Cauchy sequence must after some finite
number of terms be repetition of some integer and thus converges to that
integer.

I am having trouble explaining why this does not violate the Baire Category
Theorem. Afterall, Z is a countable union of singletons. Doesn't that
mean it is a countable union of nowhere dense (even meager) sets?

Either I don't understand "nowhere dense" or "meager" or the BCT. Hell, to
be honest, it's a healthy dose of all three.

Can anyone shed some light on this problem for me?

 - Tim 

-- 
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com