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.