Countable Complement Space is Lindelöf

Theorem

Let $T = \struct {S, \tau}$ 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 $\CC$ be an open cover of $T$.

Let $U \in \CC$ 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 $\CC$ which covers each of those points.

Hence we have a countable subcover of $T$.

So by definition $T$ is a Lindelöf space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $20$. Countable Complement Topology: $2$