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I'm using the definition (without much topology) that a set $S$ of real numbers is dense in $\mathbb{R}$ if $S \cap (a,b) \neq \varnothing$ for all $a,b \in \mathbb{R}$ and $a.

My questions:

a.) If a set $S$ is dense in $\mathbb{R}$, what can you conclude about the set $A$ that contains $S$ as a subset?

I was thinking that I need to show that as long as $cl(S) \subset A$ that this would be fine. However, topology isn't a requirement for this question and I am unsure how to do it.

b.) If 2 sets $B1$ and $B2$ are both dense in $\mathbb{R}$, what can be said about the set $B1 \cap B2$?

Here I have been thinking about comparing the rationals and irrationals. I understand that they are both dense in $\mathbb{R}$ but one is countable and the other isn't. Here they have no intersection too. But, if I considered $(-\infty, 10) \cap (0,+\infty)$ there would be a non-empty intersection.

So I am not sure what can be said about such a general question....

Thanks for any guidance!

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    For (2), the [Baire category theorem](http://en.wikipedia.org/wiki/Baire_category_theorem) says something interesting, although it may just be confusing at this point.2012-02-15

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a) $A$ must be dense, since any open interval intersects $S$, and thus intersects $A$.

b) The intersection of two dense sets could be empty. Consider, as you did, the set of rationals and the set of irrationals. The intersection could possibly be dense by taking a dense set $S$ and setting $S=B_1=B_2$. Nothing can be said in general about the intersection of two dense sets. The intersection could possibly be dense, empty, or even non-empty and non-dense (for instance the rationals and the union of the irrationals with a non-dense set $A$ of rationals).

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    That last intersection is quite imaginative! Thanks for the food for thought!2012-02-15