From: Dave Seaman Subject: Re: Heine-Borel covering theorem Date: Tue, 05 Oct 1999 17:23:15 EST Newsgroups: [missing] To: Dave Rusin Keywords: Nowhere dense In message <199910052212.RAA02891@vesuvius.math.niu.edu> you write: >In article <7tdjv0$s0o@seaman.cc.purdue.edu> you write: >>On the other hand, it is also true that for every x in C and for each >>epsilon > 0, there are points belonging to the complement of C that lie >>within epsilon of x. > >Yes > >>In other words, > >No > >>C is "nowhere dense." > >Yes. > >At least, I think I have that right. R-C is indeed dense in R (the first >statement), and C is nowhere dense (in fact it's closed and contains >no intervals). But those are not the same concept. > >A set is "nowhere dense" if its closure has empty interior, >e.g. a smooth curve in the plane is nowhere dense. > >dave Yes, you're right. I also mentioned that every point of C is an accumulation point. I considered mentioning that C is actually a perfect set (equal to its own derived set), but I didn't. That fact, together with C having empty interior, would imply that C is nowhere dense.