Uncountable Particular Point Space is not Lindelöf

Theorem
Let $T = \struct {S, \tau_p}$ be an uncountable particular point space.

Then $T$ is not a Lindelöf space.

Proof
Consider the open cover of $T$:
 * $\CC = \set {\set {x, p}: x \in S, x \ne p}$

As $S$ is uncountable, then so is $\CC$, as we can set up a bijection $\phi: S \setminus \set p \leftrightarrow \CC$:
 * $\forall x \in S \setminus \set p: \map \phi x = \set {x, p}$

Hence $\CC$ has no countable subcover.

The result follows by definition of Lindelöf space.