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.....