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Dense and Nowhere Dense Sets


Date: 04/25/99 at 21:52:47
From: Patrick
Subject: Dense and nowhere dense sets

I have been looking for good definitions of these two terms. I know 
this much: A set x is dense in Y if the closure of x equals Y. Nowhere 
dense means there are no definable open sets in the collection. I need 
better definitions than this, though. Help? 

Thanks.


Date: 04/25/99 at 23:59:37
From: Doctor Tom
Subject: Re: Dense and nowhere dense sets

Hi Patrick,

I think your definition of "nowhere dense" isn't quite right. Better
is this: "A set is nowhere dense if its closure contains no open sets 
as subsets" or something like that.

What you need is not better definitions, but a better understanding of 
what dense and nowhere dense mean. The definitons above are hard to 
understand since they apply to arbitrary topological spaces, some
of which are quite bizarre.

Start by understanding what it means on a simple space - say the real 
numbers. A set A is dense in a set B if for any element of B, we can 
find a point in A arbitrarily close to it. (This definiton is no good 
for arbitrary topological spaces, since there may not be a definition 
of distance.)

Nowhere dense means that you can't find even a tiny interval where the 
points are dense.

For example, the rational numbers are dense in the reals, since every 
real can be approximated arbitrarily closely by rationals.

Any finite set of reals is nowhere dense. In fact, if you have an 
infinite set of points that converge to a single point, that's still 
nowhere dense. Similarly with an infinite set of points that approach 
a finite number of points.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
College Analysis
High School Analysis
High School Sets

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