Countability of disjoint intervals

According this problem/solution set from an MIT class (http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/exams/exam1_sol.pdf), the assertion:

"Every collection of disjoint intervals in R is countable."

is True, because "every interval contains a rational number", and the rationals are countable.

It seems to me this should be False, with possible counterexample:

{ [x,x] | x is an element of R}

ie the set of all singelton intervals on R. Why isn't this a valid counterexample?

Your thinking is correct; the set of all singleton sets of R is certainly uncountable.

It seems that the question meant something like "Every collection of disjoint open intervals in R is countable." (In this case, the claim that each interval contains a rational number is valid.)

Maybe there was some convention in the course that "interval" meant open interval, or excluded singleton sets; perhaps it's simply a mistake. Either way, it's good that you noticed this detail!

Because "singleton interval" is usually not considered to be an interval.