arbitrary collection of disjoint open intervals is countable...

This statement is wrong. Here's a counter - example. The union of countable family of open intervals (- n , n) is R which is uncountable. By Lindelof theorem, an arbitrary union of open sets can be reduced to a countable union. Therefore, if a union of a countable family of open intervals is uncountable, then a union of an uncountable super-family of open intervals can be uncountable. QED can somebody throw light on the above paragraph.....