Proving the uncountability of $[a,b]$ and $(a,b)$ I am trying to prove that $[a,b]$ and $(a,b)$ are uncountable for $a,b\in \mathbb{R}$. I looked up Rudin and I am not too inclined to read the chapter on topology, for his proof  involves perfect sets.
Can anyone please point me to a proof of the above facts without point-set topology?
I am thinking along these lines:
$\mathbb{R}$ is uncountable.If we can show that there exists a bijection between $(a,b)$ ad $\mathbb{R}$ we can prove $(a,b)$ is uncountable.But I am not sure how to construct such a bijection. 

My strategem would be to prove firstly this more general result, whose name I'd like to ask if anybody knows? Also, is there a question on MSE about this theorem? 

Let $S$ be an infinite set such that $|S| > |\mathbb{N}|$. Also, let $C \subset S $ be countable. Then:
  $1.$ $|S - C|$ is infinite.
   $2.$  $|S| = |S - C|$

Subsequently, after you prove that $|(a,b)| = |\mathbb{R}|$, apply this theorem with $C = \{a,b\}$ and $S = [a,b]$ to prove that $|\; [a,b] \; | = |(a,b)| = |\mathbb{R}|$.
This stronger catholic result can be used frequently, such as to prove further equalities of cardinalities in addition to those above:
$|\; [a,b] \; | = |(a,b)| = |\; [a,b) \; | = |\; (a,b] \; |= |\mathbb{R}|. $
