Countable Particular Point Space is Lindelöf

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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.

$\blacksquare$


Sources