Re: a subset in X A is dense iff X intersection A is not empty



John Smith wrote:
I am reading a book about manifolds and this is something I should know but I don't know how to prove it.
Show that a subset A in X is dense if and only if every nonempty open set in X contains a point of A. 



This is true for any topological space X and dense subset A.

The proof is a simple application of the definition of a
dense subset. You do need to use both of the following facts:

	1. The complement of a closed subset is open,
	2. The only closed set containing a dense subset
	   is the whole space X.

If you'll assure me that this isn't a homework problem, I'll post
a simple proof (I've written one up for the occasion, but just
now decided I should check on the homework issue; no offense is
intended)

Dale
.