Countable Particular Point Space is Lindelöf

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

Then $T$ is a Lindelöf apace.

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

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

Hence $\mathcal C$ is its own countable subcover.

The result follows by definition of Lindelöf apace.