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!