An example of Lindelöf Theorem in Real Analysis In any elementary real analysis books Lindelöf Theorem stated as follows: "Let $C$ be a collection of open sets of real numbers. Then there is a countable sub-collection of the elements of $C$ such that the union of the elements of this sub-collection is equal to the union of the elements of $C$."
(the proof can be found almost in the same way in many real analysis book)
I am trying hard to find an example of an uncountable collection $C$ of disjoint open sets of real numbers and a countable sub-collection of $C$ such the the union of the elements of this sub-collection  equal to the union of the elements of $C$
  Your Help/Hints is much appreciated.
 A: In general, a topological space $X$ is said to be a Lindelöf space if every open cover contains a countable subcover. This is a weaker condition than that of compactness, which requires that every open cover contains a finite subcover.
To show that $\mathbb R$ has the Lindelöf property, let $\mathcal O$ be a collection of open sets such that $\bigcup_{U\in\mathcal O}U = \mathbb R$. Then for each rational number $q$, there exists $U\in\mathcal O$ with $q\in U$. Let $$U_q = \bigcup_{U\in\mathcal O,\, q\in U} U$$
for all $q\in\mathbb Q$. Then each $U_q$ is open as the union of open sets, and if $x\in\mathbb R\setminus\mathbb Q$, then $x\in U_x$ for some $U_x\in\mathcal O$, and there exists a rational number $q\in U_x$, so that $x\in U_q$. Hence $$\mathbb R = \bigcup_{q\in\mathbb Q}U_q$$
and $\{U_q : q\in\mathbb Q\}$ is a countable subcover of $\mathcal O$.
Your condition that the open cover be disjoint is irrelevant, as the above argument would still apply (even if an uncountable, disjoint open cover of $\mathbb R$ could exist, which it cannot as @Carlos Israel Jrl pointed out).
