Maximum number of disjoint intervals in the interval $(0,1)$

Question

It's obvious that if we divide this interval into infinite disjoint intervals, we can pick one rational number from each interval. But can we not also pick an irrational from each interval proving that there will be uncountable intervals?

Answer 1

No, picking an irrational from each interval gives an injective function from the set of intervals to the set of irrational numbers. So the only thing this proves is that there are at most continuum many such intervals (recall: continuum is the cardinality of the reals, and hence of the irrationals).

As you already argued, we can also pick a rational from each interval. By the same reasoning as above we then conclude that we had at most countably many intervals. This does not contradict each other, we just get a tighter bound.

In fact this last bound is the tightest possible, because $$ \{(1-1/n, 1-1/(n+1)] : n \geq 1\} $$ is a infinitely countable set of disjoint intervals covering $(0,1)$.