Countable Complement Space is Lindelöf

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Theorem

Let $T = \left({S, \tau}\right)$ be a countable complement topology on an uncountable set $S$.


Then $T$ is a Lindelöf space.


Proof

By definition, $T$ is a Lindelöf space if and only if every open cover of $S$ has a countable subcover.

Let $\mathcal C$ be an open cover of $T$.

Let $U \in \mathcal C$ be any set in $C$.

It covers all but a countable number of points of $T$.

So for each of those points we pick an element of $\mathcal C$ which covers each of those points.

Hence we have a countable subcover of $T$.

So by definition $T$ is a Lindelöf space.

$\blacksquare$


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