Re: Nowhere dense sets question..,
- From: William Elliot <marsh@xxxxxxxxxxxxxxxxxx>
- Date: Fri, 10 Jun 2005 05:26:46 -0700
On Fri, 10 Jun 2005, Lasse wrote:
>
> > So I need to show that the intersection of two dense sets is dense. But I
> > can't see how this is true. For example, let the intersection of the
> > rationals with the irrationals is empty yet they are both dense in the
> > reals.
>
> Of course the intersection of two dense sets is not necessarily dense.
> But remember that you know more about the complement of a nowhere dense
> set: it contains an *open* dense set.
>
> So if you follow your line of argument, you only need to show that the
> intersection of two OPEN dense sets is dense, which is not difficult.
>
Exercise: The intersection of an open dense set and a dense set is dense.
Thus
int cl A = nulset = int cl B ==> int cl A\/B = nulset
can be improved to
int cl A = nulset = int B ==> int((cl A) \/ B) = nulset
which implies the forme simply by setting B = cl B
.
- References:
- Nowhere dense sets question..,
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- Re: Nowhere dense sets question..,
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- Nowhere dense sets question..,
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