Uncountable Discrete Space is not Lindelöf

Theorem
Let $T = \struct {S, \tau}$ be an uncountable discrete topological space.

Then $T$ is not a Lindelöf space.

Proof
Consider the set $\CC$ of all singleton subsets of $S$:
 * $\CC := \set {\set x: x \in S}$

From Discrete Space has Open Locally Finite Cover, $\CC$ is an open cover of $S$ which is finer than any other open cover of $S$.

That is, $\CC$ is an open cover of $S$ which is uncountable and has no countable subcover.

(Note that a subcover is a refinement of a cover.)

So by definition $T$ can not be a Lindelöf space.

Also see

 * Countable Discrete Space is Lindelöf