Dense and Nowhere Dense SetsDate: 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/ |
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