Intersection of an uncountable set and an uncountable dense set

May 16, 2024, 12:08 pm

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Question :

Let $A,B\subseteq\mathbb{R}$ with $A$ being uncountable and $B$ being uncountable and dense.
Must it be true that $A\cap B≠\emptyset?$

Since $(\mathbb{R}\setminus\mathbb{Q})\cap\mathbb{Q}=\emptyset,$ we have that the intersection of an uncountable set and a countable dense set may be empty.

Since $(-\infty,0)\cap(0,\infty)=\emptyset,$ we have that the intersection of an uncountable set and an uncountable (but non-dense) set may also be empty.

So, it is clear that we need to make use of both the uncountability and the density of $B.$

I don't know where to go from here. From what I saw when I looked this up, this might require topology to show. However, I don't know any topology.

Is this statement true? Is there any elementary-ish way to see this?

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