Uncountable Discrete Space is not Lindelöf

Theorem
Let $T = \left({S, \tau}\right)$ be an uncountable discrete topological space.

Then $T$ is not a Lindelöf space.

Proof
Consider the set $\mathcal C$ of all singleton subsets of $S$:
 * $\mathcal C := \left\{{\left\{{x}\right\}: x \in S}\right\}$

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

That is, $\mathcal C$ 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