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

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.
