Is there any way to visualize what is happening here, or do we just have to rely on the definitions/theorems? 1. Every open segment of reals (a,b) is uncountable 2. Every open segment of reals contains a rational 3. cardinality(R) = cardinality(PowerSet(N)). So an uncountable set is "much" bigger. 4. The function f(x): = 1 for x rational, 0 for x irrational is discontinous everywhere. I follow the proofs/definitions justifying all the above. But intuitively, it doesn't make sense. There are so many more uncountable numbers, it seems like there must be a bunch of them adjacent without a rational in between. But there can't be that many adjacent, because then we could actually create an open segment without a rational. But as soon as you have more than 1 point in a connected metric space, then we're back to an uncountable set, and rationals creep back in. Does anyone have an intuitive explanation for this?