Intuition about open cover of $\mathbb Q$ of arbitrary small measure

Question

I have a question that is a bit vague. One can construct (more ore less explicitly) an open cover of say $\mathbb Q\cap[0,1]$ of arbitrary small Lebesgue-measure. My question asks for an explanation of how one should think about such a cover. The best would be a sort of visualisation (but this is certainly asking for too much) but my imagination is puzzled by the fact that an arbitrary big set (in a measure-theoretic sense) of [0,1] can be left uncovered while the rationals are covered and they are dense... Does anyone has a explanation of this?

Answer 1

I think there are two things here which our intuition needs to get used to:

• There are just so much more irrationals than rationals in $[0..1]$.
• These balls used to cover the rationals in $[0..1]$ in such a construction are getting really small, and yet they are all over the place.

Whenever I try to visualize the construct, my brain gets trapped in the following fallacy: If there are balls covering all the rationals in $[0..1]$, but leaving out so many irrationals, wouldn’t there be a lot of empty space between them? Couldn’t I find some rational in these empty spaces?

Also I automatically envision the balls to be disjoint and well-ordered (by their centers).

Here is my intuition about that picture: The thing is that the balls are not disjoint and well-ordered. There possibly is a lot of space between any two balls, but there are also a lot of other balls between any two balls. There is no “next” ball. And almost all of the balls in between any two balls are really, really tiny in comparison. In fact, so tiny that they only cover a small portion of the space between. They are just big enough to be called a ball. And as you zoom into this picture, you will find more and more balls popping up, but they get tinier and tinier.

At some level, pick some arbitrary set of irrationals. It’s not hard to imagine all subsequent balls that turn up by zooming in to avoid that given set of irrationals by being very small. This is what’s going on.

Answer 2

They say, in mathematics you don't understand things, you just get used to them.

Forget the cover for a moment; how would you visualize $\mathbb Q$ itself? It is so thin, it has only countably many points, and yet you can't go anywhere on a line without finding them all around you.

Answer 3

I don't think a sane persion would easily visualize such a construct. The construct will be a dense set with just isolated points cut out (since $\mathbb Q$ is dense).

The construct is to simply enumerate the rational numbers and surround them with an open set such that $m(V_n) = \epsilon k^{-n}$. Since $\bigcup V_n$ will cover all rational numbers and $m\bigcup V_n \le \sum \epsilon k^{-n} = \epsilon / (1-k)$ you can thereby cover the $\mathbb Q$ with arbitrarily small open set.