Uncountable Particular Point Space is not Lindelöf

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Theorem

Let $T = \left({S, \tau_p}\right)$ be an uncountable particular point space.


Then $T$ is not a Lindelöf space.


Proof

Consider the open cover of $T$:

$\mathcal C = \left\{{\left\{{x, p}\right\}: x \in S, x \ne p}\right\}$

As $S$ is uncountable, then so is $\mathcal C$, as we can set up a bijection $\phi: S \setminus \left\{{p}\right\} \leftrightarrow \mathcal C$:

$\forall x \in S \setminus \left\{{p}\right\}: \phi \left({x}\right) = \left\{{x, p}\right\}$

Hence $\mathcal C$ has no countable subcover.

The result follows by definition of Lindelöf space.

$\blacksquare$


Sources