We want to prove that $\displaystyle [a,b] $ and $\displaystyle (a,b) $ are uncountable.

The book says that there are bijections from $\displaystyle (a,b) $ onto $\displaystyle (-1,1) $ onto unit semicircle. From this point how do we deduce that $\displaystyle (a,b) $ is uncountable?

$\displaystyle [a,b] $ is then uncountable if we prove that $\displaystyle (a,b) $ is uncountable.